Equivalent Flow Stress Research Articles

A 90-meter Split Hopkinson Tension–Torsion Bar: Design, Construction and First Tests

AbstractThis article describes the design, construction, and first experimental results of a 90 m-long Hopkinson bar which can perform high strain rate tests in a combined tension–torsion state. The system configuration is analogous to the classic Hopkinson bar technique, consisting of three bars: a pre-stressed bar, an input bar, and an output bar; the sample is placed between the input and output bar. The measurement is also based on the classical three-wave method, where the incident, transmitted and reflected waves are measured. Incident compression and torsional waves are simultaneously generated by the failure of a fragile element that connects the pre-stressed bar to electromechanical actuators; the indirect Hopkinson tension bar technique is exploited, where the compression wave reaches the end of the output bar without stressing the sample and is reflected as a tensile input wave. The length of the bars is designed so that the tensile wave reaches the sample from the output bar side at the same time as the torsion wave comes from the input bar. Void tests were carried out first, for preliminary analysis of the system behaviour. Then, successful tests have been conducted on samples made of AA7075 T6, both in combined and pure tension–torsion states; it has been possible to measure the tension–torsion stress–strain curves, from which the equivalent flow stress–strain curve has been evaluated and compared to the quasi-static ones.

Post necking evaluation of the tensile test using artificial neural networks

This paper introduces a new method for evaluating the tensile test using finite element simulations and data-driven artificial neural networks. For this purpose, a synthetic data set was generated by finite element simulations using LS-DYNA. Artificial neural networks of two different topologies were trained and tested on parts of this synthetic data set. The networks use geometry information of the necking area as input data to predict a correction factor to convert the stress obtained from the tensile test to the equivalent flow stress. The best models are evaluated on the test set and good results are achieved.

Analytical and numerical crashworthiness uncertainty quantification of metallic thin-walled energy absorbers

Thin-walled tubular components are featured in transport structures for increased occupant protection in the event of a crash. However, when the limits of their design are sought through deterministic procedures, components often become unreliable due to uncertainties which may cause them to underperform or even fail. Seeking a deeper understanding of the effect of incertitudes on the thin-walled tubes' performance, this research focuses on the crashworthiness quantification of diverse sources of uncertainties in the progressive collapse of these components under axial loads. For a broader insight on the tubes’ behavior, diversity is considered and scrutinized on all the main fields involved in the research; studying two cross-sections, three sources of uncertainties, and the two epitomical crashworthiness metrics, namely the average and peak crushing loads. The process is undertaken via three different methods, combining analytical formulas, numerical simulations, and surrogate modeling. Results show that the best approximation is offered by the multivariate adaptive regression splines metamodel, yielding similar mean values as with the statistical propagation of the analytical formulas, while the standard deviation is overestimated by 1%–3%. The numerical noise quantified for the simulations results shows oscillation frequencies below 40% and a breadth of three-sigma under 9% for both metrics. The uncertainty quantification of both tubes offers a similar response when studying geometric uncertainties, with the plate thickness having a more relevant effect on the results than diameter or edge length. However, material uncertainties affect the absorbers in an opposite manner, as variations in the elastic modulus contribute the most to the square section metrics, while the circular tube is more affected by variations in the equivalent flow stress. Incertitudes in the operational conditions lead to reduced peak load values when the impact angle varies from a perfect axial collision, consequently delivering reduced mean values and high standard deviations.

Reverse identification of high-temperature constitutive parameters of fused silica based on orthogonal cutting theory

<p indent=0mm>Laser-assisted machining (LAM) is a new type of thermomechanical processing technology that provides high-efficiency, low-cost, and high-quality processing of fused silica and other hard, brittle materials. However, the LAM process usually involves large strains, large strain rates, and high temperatures, which are difficult to observe in experiments. Finite element cutting simulations are effective for predicting the cutting mechanism of the cutting process, but an accurate material constitutive equation must be used for an accurate simulation. To obtain more accurate simulation results, this paper proposes a method for constructing the constitutive relationship of materials based on orthogonal cutting theory to compensate for the lack of a constitutive model for fused silica. According to the Oxley cutting theory, a mathematical model of the field distribution of parameters, such as stress, strain, strain rate, and temperature, in the main shear zone was established. An orthogonal cutting test platform was established, and the equivalent flow stress was calculated according to the test results. Then, a high-temperature constitutive equation for fused silica was determined by the genetic algorithm (GA) and applied to the simulation. Finally, the cutting simulation model was verified by experiments. The results showed that the simulation can better reflect the cutting force and chip morphology during an actual cutting test, which confirms the accuracy of the constitutive parameters and the reliability of the GA identification method.

The mechanical parameters modeling of heavy steel plate snake/gradient temperature rolling with the same roll diameters

The grains in the center of the heavy steel plate can be refined by the snake/gradient temperature rolling, and the deformation penetration, the microstructure, and the properties of the steel plate will be improved. The existing rolling mechanical models are not suitable for the snake/gradient temperature rolling, so it is necessary to establish the mechanical parameters model of the snake/gradient temperature rolling to instruct production. The yield criterion of rolled material was modified based on the idea of equivalent flow stress. The element stress analyses were carried out based on the uniform normal stress and nonuniform shear stress in the vertical sides of each slab. Then the equilibrium equation of the unit pressure based on the slab method was established on this basis. The deformation region was divided into three layers (the top layer, the bottom layer, and the central layer) and maximum four zones (back slip zone, front slip zone, cross shear zone, and reverse deflection zone) according to the temperature distribution and position of the neutral point, and then the 12 zones were formed during the snake/gradient temperature rolling. The boundary conditions of the existence of the back slip zone, the front slip zone, and the cross shear zone were established according to the relationship between the threading angle and the neutral angle. The accurate mechanical parameters model of the rolling force and rolling torque of the snake/gradient temperature rolling with the same roll diameters was set up on this basis. The ANSYS software has been used in the rolling process simulation by many scholars, and the calculating precision has been verified. So the rolling processes were simulated by the ANSYS software to validate the model precision. The results show that the maximum relative deviation of the rolling force analytic model is less than 7% compared with the numerical method, and the maximum relative deviation of the rolling torque analytic model is less than 11% compared with the measured results. The mechanical parameters model can accurately predict the rolling force and rolling torque during the snake/gradient temperature rolling with the same roll diameters, so as to provide a theoretical basis for the design of rolling mill and the setup of the process parameters.

Quantitative modeling and characterization of the size effects in microscale coining process of copper

This paper investigated size effects of copper in microscale coining process. By comparing experimental results with finite element analysis (FEA), the type and contribution of size effects could be identified. We found that the deformation size effect is the most pronounced in microscale coining where grains must be introduced into relatively small die cavities through severe plastic deformation. The magnitude of the deformation size effect was determined by the relative ratio of the grain size to the die cavity, and smaller size factor produced greater effect. We also found that feature height in microscale coining was related to grain size and flow stress of the material. Therefore, size effects could be quantitatively modeled and compensated by calculating equivalent grain size and flow stress correction. After compensating for size effects, the FEA model could accurately predict material forming performance.

A nanoindentation investigation of local strain rate sensitivity in dual-phase Ti alloys

Using nanoindentation we have investigated the local strain rate sensitivity in dual-phase Ti alloys, Ti–6Al–2Sn–4Zr-xMo (x = 2 and 6), as strain rate sensitivity could be a potential factor causing cold dwell fatigue. Electron backscatter diffraction (EBSD) was used to select hard and soft grain orientations within each of the alloys. Nanoindentation based tests using the continuous stiffness measurement (CSM) method were performed with variable strain rates, on the order of 10−1 to 10−3s−1. Local strain rate sensitivity is determined using a power law linking equivalent flow stress and equivalent plastic strain rate. Analysis of residual impressions using both a scanning electron microscope (SEM) and a focused ion beam (FIB) reveals local deformation around the indents and shows that nanoindentation tested structures containing both α and β phases within individual colonies. This indicates that the indentation results are derived from averaged α/β properties. The results show that a trend of local rate sensitivity in Ti6242 and Ti6246 is strikingly different; as similar rate sensitivities are found in Ti6246 regardless of grain orientation, whilst a grain orientation dependence is observed in Ti6242. These findings are important for understanding dwell fatigue deformation modes, and the methodology demonstrated can be used for screening new alloy designs and microstructures.

Tensile Properties of Sheet Metals at Intermediate Strain Rates with Pre-Strain

This paper investigates the tensile properties of sheet metals at intermediate strain rates in relation to the pre-strain history induced by the fabrication process. Sheet metals such as SPCEN(DDQ), EZNCD(DQ), SPCC(CQ), SPRC340R, and SPRC440E were selected with the pre-strain of 2, 5, and 10% by tensile elongation with static tensile test machine, which could be equivalent to the plastic strain induced by a sheet metal forming process. The pre-strained specimens were elongated at the strain rate ranged from 1 to 100 s−1. The experimental results reveal that the stress-strain curves are noticeably influenced by the pre-strain at the intermediate strain rate for low yield stress steels such as SPCEN, EZNCD, SPCC, and SPRC340R. It is also noticed that the flow stress does not trace the given stress-strain curve when a sheet metal is pre-strained and the yield stress as well as the ultimate tensile strength increases. This behavior is against the old conjecture that the flow stress follows the stress-strain curve even with the pre-strain. This paper suggests a new flow stress model for pre-strained sheet metals. The flow stress model consists of two terms: the first term is the ratio of the yield stress of a pre-strained specimen to the equivalent flow stress obtained from the test without pre-strain; the second term is the general empirical flow stress model without the pre-strain effect.

Superplastic Forming of Friction Stir Welded AA6061-T6 Alloy Sheet with Various Tool Rotation Speed

This study attempts to investigate the superplastic forming (SPF) of friction stir welded (FSW) AA6061-T6 alloy sheet at various tool rotation speeds in the range of 500 to 2000 rpm. The effect of FSW on SPF free blow forming of AA6061-T6, pole height, pole thickness, equivalent strain rate, and equivalent flow stress were investigated at constant pressure and constant temperature. Using the Cheng model the pole thickness, the equivalent strain rate, and equivalent flow stress were determined from superplastic free blow forming experiment. The finite element modeling and simulation is performed over the SPF of FSW specimens using selective superplasticity method. Experimental results indicate that tool rotation speed is the critical parameter during friction stir welding that has a greater influence on SPF. The theoretical modeling results exhibit that the SPF of friction stir welding can be practically applied to determine pole thickness, strain rate, flow stress, and strain rate sensitivity index. The finite element modeling results were found to be fairly agreeing with the experimental results. Hence, superplasticity can be significantly enhanced by friction stir welding by varying the FSW tool rotation speed.

Mathematical Modelling to Evaluate the Superplastic Material Constants by Bulge Test

The mechanical behaviour of a superplastic material is often modelled by the power law relationship between the equivalent flow stress, the equivalent strain and the equivalent strain-rate at least over a limited range of strain rates. This paper introduces an original mathematical modelling to determine the superplastic material constants m, n and K by means of experimental tests carried out using a standard forming die geometry.

Correlation of particle impact conditions with bonding, nanocrystal formation and mechanical properties in kinetic sprayed nickel

Owing to the specific high-strain-rate thermomechanical characteristics of Ni particle impact in kinetic spraying, the rebound phenomenon of the impacting particles hinders the formation of the first layer and impedes successful build-up of the coating. Even at higher impact velocities, the deposition efficiency of the coating is quite low because of excessive kinetic energy, which induces the rebound and/or erosion of the highly flattened particles. This paper reports noticeably improved bonding and deposition characteristics of Ni particles resulting from suppressed equivalent (von Mises) flow stress and enhanced interface heat-up as a result of powder preheating. Experimental observations coupled with finite-element modeling (FEM) corroborate the fact that the thermally softened Ni particle is very effective for enhanced adhesive and cohesive bonding. Based on the FEM results, the thermal boost-up zone, increased by thermally accelerated adiabatic shear instability, is proposed as a crucial factor for enhancing bonding between the particles, which is essential in producing better coating properties. Moreover, nanocrystal formation (<100nm) in the coating was more pronounced than cases previously reported in the literature, mainly because of the enhanced thermal activation and straining of the severely deformed particles, which was verified by transmission electron microscopy investigations and nanoindentation tests.

AZ31 Magnesium Alloy Parameters Identification through Inverse Analysis at 713 K

This paper introduces a fast and accurate procedure for determining the constants of magnesium AZ31 alloy at 713 K. The material behaviour is modelled by means of the power law relationship between the equivalent flow stress, the equivalent strain and the equivalent strain-rate within a narrow equivalent strain-rate range. Bulging tests were carried out in isothermal conditions (713 K) and at constant pressure in order to determine the material constants. It is necessary to evaluate the displacement and the thickness evolutions at the dome apex of the metal sheet. The time-displacement curve was obtained by laser measurements whereas a large number of bulging tests, interrupted at preset time intervals, were carried out to evaluate the thickness. The thickness was measured directly using a two-digit micrometer. The material constants, m, n and K were obtained in the power law relationship by means of constant pressure bulging tests coupled with the use of an inverse analysis technique. The results of comparison between experimental and numerical tests are shown and they indicate that the material constants can be accurately evaluated.

Determination of Uniaxial Flow Stress Curve Using Aero-Bulge Test for Very Thin Copper Sheet

Determination of the flow stress curve is an important step for precisely describing material behavior in Finite Element simulations. The flow stress curve is generally determined by taking a uniaxial tensile test as a standard. In the case of very thin sheet, since the fracture is generated at a low strain, there is not enough uniaxial data obtained to be applied in the FE simulation. The reason for this is that charactering plastic deformation at a large strain values by extrapolating a flow stress curve which is based on insufficient measurement data is highly susceptible to error. Bulge test is useful method for determining the equivalent biaxial flow stress curve up to a large strain. In this paper, the biaxial flow stresses curve for very thin copper sheet with thickness 35 and 50 μm were determined using the aero-bulge test. A new empirical model was derived for the estimation of the sheet thickness at the pole. After the compatibility between uniaxial and biaxial flow stresses was verified, the uniaxial flow stress curve was determined from the aero-bulge test using reverse engineering. The methodology of extrapolation of the flow stress curve at a large strain was finally proposed for application in FE simulations.

New generalised plasticity equation for compressible powder metallurgy materials: a new parallel RK-Butcher method

In this paper, we propose a highly non-linear mathematical expression for characterising the flow behaviour of partially dense P/M materials during the case of simple upsetting – compression test, taking functional dependence of strain hardening index on a matrix strain and yield matrix (of partially dense material) on a mixed combination of matrix strain and strain sensitivity into account. Equivalent flow stress of matrix is considered to be a function of relative density, matrix strain and strain rate sensitivity parameters. A new parallel Runge-Kutta-fifth order algorithm with adaptive step-size control has been proposed for numerical integration of the resultant mathematical expression to compute the value of yield stress of partially dense P/M porous materials. Furthermore, a modest effort has been made to bring out the effectiveness of the adaptive step-size control. Numerical results show that the proposed parallel Runge-Kutta-fifth order algorithm with the adaptive step-size control increases the computational speed by more than eight times in an interval when solving the highly non-linear prevailing equation in comparison with the above algorithm with the fixed constant step-size.

Effect of high strain rates on peak stress in a Zr-based bulk metallic glass

The mechanical behavior of Zr41.25Ti13.75Cu12.5Ni10Be22.5 (LM-1) has been extensively characterized under quasistatic loading conditions; however, its mechanical behavior under dynamic loading conditions is currently not well understood. A Split–Hopkinson pressure bar (SHPB) and a single-stage gas gun are employed to characterize the mechanical behavior of LM-1 in the strain-rate regime of 102–105/s. The SHPB experiments are conducted with a tapered insert design to mitigate the effects of stress concentrations and preferential failure at the specimen-insert interface. The higher strain-rate plate-impact compression-and-shear experiments are conducted by impacting a thick tungsten carbide (WC) flyer plate with a sandwich sample comprising a thin bulk metallic glass specimen between two thicker WC target plates. Specimens employed in the SHPB experiments failed in the gage-section at a peak stress of approximately 1.8 GPa. Specimens in the high strain-rate plate-impact experiments exhibited a flow stress in shear of approximately 0.9 GPa, regardless of the shear strain-rate. The flow stress under the plate-impact conditions was converted to an equivalent flow stress under uniaxial compression by assuming a von Mises–like material behavior and accounting for the plane strain conditions. The results of these experiments, when compared to the previous work conducted at quasistatic loading rates, indicate that the peak stress of LM-1 is essentially strain rate independent over the strain-rate range up to 105/s.

New constitutive equation for plasticity in high speed torsion tests of metals

Present work investigates a phenomenological plasticity equation for the plastic behavior of metals deformed in high speed torsion tests. The tests were carried out at room temperature in a laboratory torsion test equipment and also in an universal tensile test machine on annealed commercial pure copper and aluminum specimens. The tensile tests were performed at room temperature by an universal testing machine at low strain rate of 0.017/s. The experimental torsion tests were carried out at constant angular speed that imposed a constant shear strain rate to the specimen. In the tests, the rotation speed were set to 61 rpm and 200 rpm which imposed high strain rates of about 2/s and 6.5/s respectively in tubular specimens. The torsion test lasted between 0.5 s and 1 s. The experimental hardening curves of equivalent stress versus strain in torsion were sigmoidal type and were fitted by different constitutive equations for plasticity that takes into account the strain hardening and thermal softening effects due to local temperature increase as the Johnson-Cook model, the modified Voce equation and others. The specimen local temperature increase was calculated assuming adiabatic deformation process. It is proposed a new constitutive equation or modified Voce equation for the equivalent flow stress which considers the effects of strain hardening, strain rate hardening and thermal softening for the best fit to the present experimental data of annealed aluminum and copper.

Hot Workability and Constitutive Equations of ZM21 Magnesium Alloy

One of the most challenging activities for magnesium industry is to increase productivity by introducing cost-effective processes. While die-cast magnesium alloys are in worldwide use, even in the demanding sector of automotive industry, where cost is the major asset, wrought magnesium alloys still not widely available. The aim of the present study was to evaluate the high temperature response by torsion testing of a ZM21 alloy, to calculate the constitutive equations to be used in FEM simulation of the extrusion process. The ZM21 alloy that was considered in the present study was Direct-Chill casted by Alubin Ltd., Israel. Torsion tests were carried out in air on a computer-controlled torsion machine, under strain rates ranging from 10-2 to 5 s-1 and temperatures from 200 to 400°C. The equivalent peak flow stress was related to temperature and strain rate by means of the conventional power-law and by the sinh equation. The microstructure of the alloy, even at 300°C, appeared largely unrecrystallized, with elongated grains; also at 400°C, the structure is more equiaxed, but elongated structures still appears.

Critical examination of strain-rate sensitivity measurement by nanoindentation methods: Application to severely deformed niobium

Two different techniques for measuring the strain-rate sensitivity (SRS) exponent, m, of the power law relating the equivalent flow stress and the equivalent plastic strain rate are analyzed, paying special attention to the errors induced by the thermal drift rate and by the indentation-size effect. It is shown that the resolution and accuracy of indentation-creep tests are affected by these factors. An alternative method of m determination, the rate-jump test, avoids the influence of both the thermal drift and the indentation-size effect. Both methods are applied to the SRS determination of pure niobium samples with different plastic strain levels imparted by equal channel angular pressing. The results show the advantages of the rate-jump test. The observed strong dependence of the room temperature SRS of niobium on its dislocation density and on the strain rate are satisfactorily explained by the relative contributions of the Peierls stress and the dislocation density hardening to the flow stress of the samples.

Metallographic analysis of the deformation microstructure of copper subjected to orthogonal cutting

Metallographic sections taken from commercially pure copper samples, which formed continuous chips during the orthogonal cutting process, were examined to determine the displacement gradients (or the shear angles), and the local strain gradients generated in the material ahead of the cutting tool tip. The local flow stress variations in the deformed material ahead of the cutting tool tip were estimated from the microhardness measurements. An empirical relationship was established between the equivalent stresses and strains in the deformed material ahead of the tool tip. The equivalent flow stress and the equivalent strain relationship obeyed a Voce type exponential equation. It was shown that a saturation flow stress (422 MPa) is reached at strain levels ε ̄ >3.0. Plastic deformation work expended for each strain increment and the corresponding local temperature increase were calculated from the empirical flow curve. The highest temperatures rise (857 °C) was estimated to occur immediately ahead of the tool tip, and in the secondary deformation zone adjacent to the tool rake face.

Multiaxial Experiments and Constitutive Modeling of Superplasticity

In the first part of the paper, multiaxial experiments of 5083Al alloy are carried out at 833K at constant strain-rates to elucidate the characteristic features of superplastic deformation under general forming conditions. Monotonic tension, compression and torsion tests were performed by using solid circular cylindrical specimens together with thin-walled tubular specimens. The values of the equivalent strain-rate of von Mises type are specified as 2 × 10 -3 , 1 × 10 -3 , 5 × 10 -4 and 2 × 10 -4 s -1 . The significant strain-rate dependence of flow stress and the material softening due to cavity growth are observed under the tension tests. The results of the tension and compression tests show that the material hardens under tension while it softens under compression. The comparison between tension and torsion tests shows that the equivalent flow stress of von Mises type under torsion is much smaller than that of the tension at the same equivalent strain-rate of von Mises type. Based on these observations, a constitutive model of superplasticity is then formulated based on the finite deformation theory. The rate of deformation tensor is divided into the sum of the elastic and inelastic parts, and the former part is represented by the isotropic linear elastic law. The inelastic part is modeled based on the invariant theory by employing the steady-state creep law of hyperbolic sine type and by taking account of the effects of yielding, grain and cavity growth. The rate of drag stress is assumed to be the sum of the terms due to the grain and cavity growth. The evolution equation of the grain growth is formulated by taking account of static growth and strain-induced growth. The evolution equation of the cavity volume fraction is represented by the sum of the two terms due to cavity diffusion and viscoplastic deformation. Comparison of the results of the proposed model with those of the experiments shows that the proposed model gives satisfactory predictions.