Bridgman Correction Research Articles

Cyclic true stress true strain relation in the large strain region of common steel for welded structures

Cyclic true stress - true strain curves were obtained using general-purpose shipbuilding steel in the large strain region. Test methods are incremental step test and static tensile test after cyclic loading. The stress–strain curves for static and cyclic loading were compared to investigate the cause of the difference between each other. A simple method for estimating the cyclic stress-strain curves was proposed. The diameter and curvature of the smallest cross-section of the specimen were measured using telecentric measurement device, and the true stress - true strain curve was obtained using the Bridgman correction method. Using the measurement method, cyclic stress-strain diagrams were obtained in the region where the true strain exceeded 1% by performing cyclic tensile tests using the incremental step method and static tensile tests after cyclic loading. The influence of the test method and maximum displacement conditions on the cyclic stress-strain curve was small within the scope of this study. There was little difference in the elongation between the static tensile test and the static tensile test after cyclic loading. The difference in yield stress between static and cyclic loading was discussed in terms of macro-yield mechanisms at intergranular and transgranular. A simple method for estimating cyclic stress-strain curves from a static stress-strain curve was proposed. The specimens used in this study are general-purpose shipbuilding steels, and the results should be applicable to similar steels for welding and structures.

A new analytical model to predict the formation of necking instabilities in porous plates subjected to dynamic biaxial loading

In this paper, we have developed a linear stability analysis to predict the formation of necking instabilities in porous ductile plates subjected to dynamic biaxial stretching. The mechanical behavior of the material is described with the Gurson–Tvergaard–Needleman constitutive relation for progressively cavitating solids (Gurson 1977; Tvergaard 1981, 1982; Tvergaard and Needleman 1984) which considers the voids to be spherical and the matrix material isotropic with yielding defined by the von Mises (1928) criterion. The analytical model is formulated in a two-dimensional framework in which the multiaxial stress state that develops inside the necked region is approximated with the Bridgman (1952) correction factor, superimposing a hydrostatic stress state to the uniform stress field that develops in the plate before localization. As opposed to the linear stability models published so far to model dynamic necking in ductile plates, which consider the material to be fully dense and incompressible, the approach developed in this paper provides new insights into the interplay between porosity and inertia on plastic localization. In addition, the predictions of the theoretical model for the critical strain leading to necking formation have been compared with unit-cell finite element calculations performed in ABAQUS/Explicit (2019). Satisfactory quantitative and qualitative agreement has been found between the theoretical and the computational approach for loading paths ranging from plane strain tension to nearly equibiaxial tension, loading rates varying from 100 to \(10{,}000~\text {s}^{-1}\), and different values of the initial void volume fraction ranging from 0.01 to 0.1. Both analytical and finite element results suggest that the influence of porosity on necking localization increases, due to voids coalescence, as the loading rate increases and the loading path approaches equibiaxial tension. The original formulation developed in this paper serves as a basis for analytically modeling the dynamic formability of porous ductile plates, and it can be readily extended to consider more complex porous plasticity theories, e.g. constitutive models which consider the anisotropy of the material (Benzerga and Besson 2001) and/or voids with different shapes (Gologanu et al. 1993; Monchiet et al. 2008).

Measuring true stress–strain curves of cylindrical bar samples with mirror‐assisted multi‐view digital image correlation

AbstractThis paper proposes to measure true stress–strain curves at the necking region of cylindrical bar samples by using an advanced mirror‐assisted multi‐view digital image correlation (MV–DIC) technique. The mirror‐assisted MV–DIC can measure the panoramic shape and deformation fields of a cylindrical bar sample with only a single two‐camera stereo‐DIC system and two planar mirrors. The measured panoramic shape enables accurate determination of the cross‐section area and radius of curvature at the necking region by using cylindrical or torus fitting. The retrieved cross‐section area and radius of curvature at the necking region further enable the measurement of the true stress–strain curves and even the Bridgman's correction of the true stress.

Extension of 1D linear stability analysis based on the Bridgman assumption. Applications to the dynamic stretching of a plate and expansion of a ring

The analysis of multiple necking phenomenon during dynamic stretching of a metallic plate is investigated. For that purpose, a new 1D linear stability analysis is proposed to capture the perturbation evolution. The multidimensional aspect of the stress field within the neck region is taken into account with a Bridgman correction factor. The novelty of the 1D approach is to let the time evolution of the perturbation determined by the linearized field equations. So, no predefined time dependency is assumed as in classical 1D linear stability analyses of the literature, see Zhou et al., An elastic-visco-plastic analysis of ductile expanding ring, Int. J. Impact Eng., 2006. The proposed model, named hereafter 1D-XLSA (standing for 1D-eXtended Linear Stability Analysis), can also be viewed as the restriction to 1D of the 2D-XLSA model developed in Xavier et al., Extension of linear stability analysis for the dynamic stretching of plates: Spatio-temporal evolution of the perturbation, European Journal of Mechanics-A/Solids, 2020. A comparison for a thermo-viscoplastic material with strain hardening is proposed for three possible routes: a 1D model based on the frozen coefficient theory named 1D-CLSA (standing for 1D-Classical Linear Stability Analysis), the proposed 1D-XLSA and 2D-XLSA models. It is shown that while growth rates in late deformation stage are similar, a strong difference in amplitude exists due to large discrepancies in the early deformation stage. The comparison also illustrates the importance of the modeling of multiaxiality of the stress field within the neck region and of initial defects. While differences between models are limited for perturbations with small wavenumbers, large discrepancies are observed for perturbations with large wavenumbers. Our model is also derived for the case of the extension of a cylindrical bar which is representative of the ring expansion during dynamic loading.

Experimental measurement of temperature-dependent equivalent stress-strain curves of a 420 MPa structural steel with axisymmetric notched tensile specimens

Recently, the authors in this paper proposed a correction function to determine material's equivalent stress-strain curve with axisymmetric-notched tensile specimens. In this study, tensile tests were performed at room temperature, −30 °C and − 60 °C with axisymmetric notched tensile specimens to verify this method and to identify the equivalent stress-strain curves of a 420 MPa structural steel. A high-speed camera was used together with the so-called edge-tracing method to calculate average true strain. The material's equivalent stress-strain curve was also measured with extensometer and smooth round bar specimens. Experimental results show that equivalent stress-strain curve of this structural steel is sensitive to test temperature. Equivalent stress-stress curves obtained from axisymmetric notched tensile specimens by using the proposed correction function show good agreement with those from extensometer before diffuse necking and from Bridgman correction at large strain using smooth tensile specimens. Since fracture strain strongly depends on the notch geometry, it is recommended to use axisymmetric notched tensile specimens with smaller a0/R0 when applying the proposed correction function to measure material's equivalent stress-strain curve.

A method for determining material's equivalent stress-strain curve with any axisymmetric notched tensile specimens without Bridgman correction

Large deformation analyses of problems such as plastic forming, ductile fracture with finite element method need a full range of material's equivalent stress-strain curve or flow stress-strain curve. The equivalent stress-strain curve determined from the smooth round bar specimen should be corrected after diffuse necking, since tri-axial stress state occurs in the neck. The well-known Bridgman correction method is a candidate, however, it is not accurate as the strain increases. Furthermore, it is impossible to measure the equivalent stress-strain curve of each individual material zone in a weldment with cross weld tensile tests. To cope with these challenges, a correction function and an associated test procedure are proposed in this study. With the proposed procedure, the true stress-strain curve from any axisymmetric notched tensile specimen can be converted to the material's equivalent stress-strain curve accurately and no Bridgman correction is needed. The proposed procedure can be applied to both perfectly plastic and strain hardening materials. The equivalent stress-strain curve of each individual material zone in a weldment can also be measured with the proposed procedure.

A special notched tensile specimen to determine the flow stress-strain curve of hardening materials without applying the Bridgman correction

Structural integrity assessment of weldments requires the input of flow stress-strain curve of each individual material zone. To cope with these challenges, a cylindrical cross weld tensile specimen with a notch located either in the weld metal, base metal or possibly heat affected zone has been previously developed by the authors to determine the true stress-strain curve for the material zone of interest. The disadvantage of this notched tensile testing method as well as the standard tensile testing method using a smooth specimen, is that the well-known Bridgman correction still has to be applied in order to obtain material’s equivalent or flow stress-strain curves. In this study, tensile specimens with various notch geometries have been scrutinized and a ‘magic’ specimen with a special notch geometry has been identified. By using this special notched tensile specimen, material’s flow stress-strain curve can be directly calculated from the recorded load versus diameter reduction curve and no Bridgman correction is needed. The method is very accurate for power-law hardening materials and becomes less accurate for materials with significant Lüders plateau in the initial yield region.

High cycle fatigue behavior of Al1070 alloy severely deformed by equal channel angular pressing process

In the present work, equal channel angular pressing of commercial pure aluminum 1070 was performed up to 4 passes using route Bc. For equal channel angular pressing operation, a suitable die set was designed and manufactured. X-ray diffraction analysis was used to determine the microstructure of the equal channel angular pressing-ed material. The fracture surface morphology and microstructure after fatigue were investigated by scanning electron microscopy. Mechanical properties of the equal channel angular pressing-ed material were evaluated by hardness and tension tests. Also, cyclic deformation behavior of severe plastic deformation Al1070 has been studied and results show a significant variation in hardness, ultimate strength and fatigue properties in high cycle fatigue life. Coefficient of fatigue strength σ′f and Bridgman correction factor have been obtained by S-N curve and tension test specimens, respectively, and compared before and after equal channel angular pressing process. Also an useful relation has been derived between fatigue life ( Nf) and stress amplitude ( σa) in high cycle fatigue region. Results indicated that there was not clear relation between fatigue strength coefficient and true corrected fracture stress in this case.

A physically-based constitutive model applied to AA6082 aluminium alloy at large strains, high strain rates and elevated temperatures

This paper presents a physically-based constitutive model applied to an AA6082 aluminium alloy subjected to large strains, high strain rates and elevated temperatures. The model accounts for thermo-elasticity, thermo-viscoplasticity, and strain-rate and temperature dependent work hardening, using the dislocation density as an internal variable without relating it to a detailed characterization of the microstructure evolution. The parameter identification is based on previously reported experimental results from quasi-static and dynamic tensile tests performed at temperatures from 20 to 350°C. The equivalent stress-strain curves were determined beyond necking by monitoring the cylindrical specimens with a digital camera and applying the Bridgman correction to account for the triaxial stress state. The results revealed a distinct increase of the work hardening at high strain rates, indicating a change of strengthening mechanism. A new formulation for the dynamic recovery provided reasonable predictions over the considered range of strains, strain rates and temperatures. Finite element simulations of the quasi-static and dynamic tensile tests were made to assess the accuracy of the Bridgman correction. The simulations reproduced the complex behaviour observed in the tests and validated the constitutive model and the associated parameter identification procedures in the entire range of strains, strain rates and temperatures investigated.

Work-hardening and ductility of artificially aged AA6060 aluminium alloy

The work-hardening and ductility of an artificially aged AA6060 aluminium alloy were studied based on tensile tests of smooth and notched cylindrical samples. The alloy was tested after three processing steps, each followed by artificial aging. These processing steps were casting and homogenization, extrusion, and cold rolling and heat treatment to obtain a recrystallized grain structure. Subsequent to each of these processing steps, the material was tested after artificial aging to underaged, peak aged and overaged conditions. The true stress–strain curve to failure was determined by use of a laser-based measurement system. The Bridgman correction was applied to estimate the equivalent stress–strain curves, and the work-hardening behaviour was analysed using an extended Voce approach. Fractography was applied to study the failure mechanisms for material exposed to the different processing steps and temper treatments. To evaluate the use of the Bridgman correction and to study the notch strengthening effect observed experimentally, finite element simulations were performed using the Gurson model. The experimental study demonstrates the effects of thermomechanical processing and artificial aging on the stress–strain behaviour and the tensile failure strain of the alloy.

Dynamic Tensile Characterization of Vascomax® Maraging C250 and C300 Alloys

Vascomax® maraging C250 and C300 alloys were dynamically characterized in tension with Kolsky tension bar techniques. Compared with conventional Kolsky tension bar experiments, a pair of lock nuts was used to minimize the pseudo stress peak and a laser system was applied to directly measure the specimen displacement. Dynamic engineering stress–strain curves of the C250 and C300 alloys were obtained in tension at 1000 and 3000 s−1. The dynamic yield strengths for both alloys were similar, but significantly higher than those obtained from quasi-static indentation tests. Both alloys exhibited insignificant strain-rate effect on dynamic yield strength. The C300 alloy showed approximately 10 % higher in yield strength than the C250 alloy at the same strain rates. Necking was observed in both alloys right after yield. The Bridgman correction was applied to calculate the true stress and strain at failure for both alloys. The true failure stress showed a modest strain rate effect for both alloys but no significant difference between the two alloys at the same strain rate. The C250 alloy was more ductile than the C300 alloy under dynamic loading.

Investigation of tension and compression behavior of AZ80 magnesium alloy

Magnesium alloy deformation should be carried out at high temperature and it is essential to investigate the deformation behavior of these alloys at high temperature. In this paper, practical tests are conducted on AZ80 alloy which includes tension and compression tests at high temperature and different strain rates. As this alloy is sensitive to temperature and strain rate, tension tests are difficult to carry out. The traditional compression test should be conducted in zero-friction condition but such a condition is impossible to prepare. Therefore, bulge correction factor and numerical correction factor are used to eliminate the friction effect which exists between the surfaces. Tensile tests are also carried out on the standard specimens at high temperature and different strain rates. The effect of necking phenomenon is corrected using Bridgman correction factor and is simulated using finite element software. Also T-shape compression test is used to valuate friction parameter at high temperatures.

Analysis of the Bridgman Procedure to Characterize the Mechanical Behavior of Materials in the Tensile Test: Experiments and Simulation

This note presents an experimental analysis and a numerical simulation of the mechanical behavior experienced by cylindrical specimens of pure copper during the tensile test. A set of experiments has been carried out in order to derive the hardening parameters that characterize the material response. The simulation of the deformation process during the whole test is performed with a finite element large strain elastoplasticity-based formulation. The results of the simulation show that the mechanical characterization involving the classical Bridgman correction factor, defined in terms of logarithmic strains and aimed at predicting the stress distribution at the necking zone, cannot properly describe the hardening response for this material. Therefore, the use of a different correction factor, which consequently leads to another set of hardening parameters, is proposed. Finally, an adequate experimental validation of the numerical results is obtained for this last case.

An analysis of the nonisothermal tensile test

An analysis of the uniaxial tensile test which includes the effects of heat generation and transfer on flow behavior is presented. The analysis is based on the solution of modified versions of the onedimensional force equilibrium and heat conduction equations, subject to the appropriate boundary conditions. The stress biaxiality or triaxiality developed during necking is taken into account by including a Bridgman correction in the equilibrium expression. In order to obtain solutions of the coupled deformation-heat transfer problem, the governing equations were discretized to enable numerical calculations using a finite-difference scheme. The accuracy of the analysis was confirmed by comparison of model predictions with experimental data from nonisothermal tensile tests on 1008 aluminum-killed sheet steel. These and other simulations lead to the conclusion that temperature effects are most important following the onset of necking. The development of temperature gradients during necking can substantially decrease the tensile elongation and mitigate the stabilizing influence of positive strain rate sensitivity.

A simplified numerical analysis of the sheet tensile test

A simplified numerical analysis of the tensile test for sheet metal specimens is presented with particular emphasis on the relationship between plastic flow parameters and tensile ductility. This analysis is based on a one-dimensional, “long wavelength” approximation in which the stress state is assumed to be uniaxial throughout the deformation. To account for the influence of necking and triaxiality on the flow behavior, however, an extended Bridgman correction for sheet specimens deformed under conditions of plane stress is employed. The governing equilibrium equation and boundary conditions are discretized to obtain computer solutions. Engineering stress-strain curves and strain distributions in deformed tensile specimens are compared to results from a more complex formulation previously published in the technical literature and show good agreement with it.