Continuous, large strain, tension/compression testing of sheet material

Modeling sheet metal forming operations requires understanding of the plastic behavior of sheet alloys along non-proportional strain paths. Measurement of hardening under reversed uniaxial loading is of particular interest because of its simplicity of interpretation and its application to material elements drawn over a die radius. However, the compressive strain range attainable with conventional tests of this type is severely limited by buckling. A new method has been developed and optimized employing a simple device, a special specimen geometry, and corrections for friction and off-axis loading. Continuous strain reversal tests have been carried out to compressive strains greater than 0.20 following the guidelines provided for optimizing the test. The breadth of application of the technique has been demonstrated by preliminary tests to reveal the nature of the Bauschinger effect, room-temperature creep, and anelasticity after strain reversals in commercial sheet alloys.     

A non-associated constitutive model with mixed iso-kinematic hardening for finite element simulation of sheet metal forming

In this paper an anisotropic material model based on non-associated flow rule and mixed isotropic–kinematic hardening was developed and implemented into a user-defined material (UMAT) subroutine for the commercial finite element code ABAQUS. Both yield function and plastic potential were defined in the form of Hill’s [Hill, R., 1948. A theory of the yielding and plastic flow of anisotropic metals. Proc. R. Soc. Lond. A 193, 281–297] quadratic anisotropic function, where the coefficients for the yield function were determined from the yield stresses in different material orientations, and those of the plastic potential were determined from the r-values in different directions. Isotropic hardening follows a nonlinear behavior, generally in the power law form for most grades of steel and the exponential law form for aluminum alloys. Also, a kinematic hardening law was implemented to account for cyclic loading effects. The evolution of the backstress tensor was modeled based on the nonlinear kinematic hardening theory (Armstrong–Frederick formulation). Computational plasticity equations were then formulated by using a return-mapping algorithm to integrate the stress over each time increment. Either explicit or implicit time integration schemes can be used for this model. Finally, the implemented material model was utilized to simulate two sheet metal forming processes: the cup drawing of AA2090-T3, and the springback of the channel drawing of two sheet materials (DP600 and AA6022-T43). Experimental cyclic shear tests were carried out in order to determine the cyclic stress–strain behavior and the Bauschinger ratio. The in-plane anisotropy (r-value and yield stress directionalities) of these sheet materials was also compared with the results of numerical simulations using the non-associated model. These results showed that this non-associated, mixed hardening model significantly improves the prediction of earing in the cup drawing process and the prediction of springback in the sidewall of drawn channel sections, even when a simple quadratic constitutive model is used.     

Crack growth in bi-material laminates

Growth rates of fatigue cracks have been measured in laminates fabricated by adhesively bonding layers of 2024-T351 and 7075-T6 aluminum alloys. The laminates had either two or four layers, with equal thicknesses and numbers of layers of each alloy. Fatigue-crack-propagation tests were performed with through-cracks, giving a crack-divider geometry, the results being compared to those for the two alloys tested in monolithic form. Crack-propagation rates in the bi-material laminates were intermediate between those of the monolithic alloys, with the slower growth in 2024-T351 tending to dominate over a portion of the growth-rate range. Fracture toughnesses of the laminates are also discussed.     

A hybrid membrane/shell method for calculating springback of anisotropic sheet metals undergoing axisymmetric loading

To reduce the computation cost of finite element analyses aiding die design for sheet metal stamping, a hybrid membrane/shell method was developed to determine the springback of anisotropic sheet metal undergoing axisymmetric loading. The hybrid membrane/shell method uses a membrane model to analyze the stamping operation. The bending/unbending strains and stresses varying through thickness are calculated analytically from the incremental shape determined by the membrane analysis. These bending strains and stresses and the final membrane shape are used with a shell finite element model to unload the sheet and calculate springback. The accuracy of the springback prediction with the hybrid method was verified against the springback of 2036-T4 aluminum and a DQAK steel sheet drawn into a cup. It was found that, in comparison with a full shell model, a minimum of 50% CPU time saving and a comparable accuracy was achieved when the hybrid method was used to predict springback.     

Tensile testing of materials at high rates of strain

A tension version of the split Hopkinson bar or Kolsky apparatus is developed for conducting tests in tension at high rates of strain up to 10³ s^?1. A number of aluminum, titanium, and steel alloys tested in tension show increasing degrees of rate sensitivity above 10 to 10² s^?1. Tests on 6061-T651 and 7075-T6 aluminum show measurable strain-rate sensitivity in tension at the highest strain rates, although similar tests in compression in the literature show essentially no strain-rate sensitivity. Details of the apparatus and instrumentation and guidelines for its use are presented.     

Viscoplastic constitutive modeling with one scalar state variable

State variables have been used to represent the material resistance to plastic deformation in the recent development of the viscoplastic constitutive equations. In a previous paper, an experimental method was suggested to identify the relative roles to be played by the scalar state variable (drag stress) and the tensorial state variable (back stress) in the state-variable based constitutive equations. The results on 2618-T61 aluminum alloy tested at 200°C suggested that the scalar state variable alone should be enough to model the experimental results of 2618-T61 aluminum alloy.

In the current work, an early version of the viscoplastic constitutive equation proposed by Bodner and Partom, which was formulated with one scalar state variable, was adopted to model the experimental results of 2618-T61 aluminum. Experiments included creep tests under stepwise loadings, controlled-strain-rate tests, and creep tests under nonproportional loadings. A constitutive equation based on strain hardening approach, which was developed in an earlier work, was also studied.

In order to improve the results of the Bodner and Partom's model, a recovery term which was an explicit function of the time exposed to the test temperature was suggested for the scalar state variable. Aging was discussed as one possible softening mechanism for the current material.     

Anticlastic curvature in draw-bend springback

Draw-bend springback shows a sudden decline as the applied sheet tension approaches the force to yield the strip. This phenomenon coincides with the appearance of persistent anticlastic curvature, which develops during the forming operation and is maintained during unloading under certain test conditions. In order to understand the mechanics of persistent anticlastic curvature and its dependence on forming conditions, aluminum sheet strips of widths ranging from 12 to 50 mm were draw-bend tested with various sheet tensions and tool radii. Finite element simulations were also carried out, and the simulated and measured springback angle and anticlastic curvature were compared. Analytical methods based on large deformation bending theory for elastic plates were employed to understand the occurrence and persistence of the anticlastic curvature. The results showed that the final shape of a specimen cross-section is determined by a dimensionless parameter, which is a function of sheet width, thickness and radius of the primary curvature in the curled region of an unloaded sample. When the normalized sheet tension approaches 1, this parameter rapidly decreases, and significant anticlastic deflection is retained after unloading. The retained anticlastic curvature greatly increases the moment of inertia for bending, and thus reduces springback angle.     

Anisotropic yield functions with plastic-strain-induced anisotropy

In most anisotropic yield functions, the stress exponent, M, associated with the shape of the yield surface is usually independent of plastic-strain accumulation. This does not allow for different work-hardening characteristics under various strain states, as has been observed in aluminum alloys. Assuming coefficients characterizing anisotropy do not change with plastic deformation, the M value should vary with plastic strain, relaxing the isotropic hardening assumption. To verify this, plane-strain tests along with numerical analysis were carried out with 2008-T4 aluminium and 70/30 brass. The effective stress and effective plastic-strain curve assuming plane strain and plane stress was fit to the corresponding stress-strain data obtained in uniaxial tension. This was done by allowing M value to vary with effective plastic-strain. Hill's 1979 (case iv),Hosford's 1979 and Barlat's 1991 (6 component) yield functions were evaluated. Results showed that, with all the yield functions tested, the aluminum exhibited substantial variation of M value especially at larger strains while the brass showed minor change. Relevant numerical analysis indicated that, even though all the yield functions showed noticeable changes of M as strain increases in order for the plane-strain curve to match with the uniaxial curve, this variation of M will not affect much to the prediction with Hosford's and Barlat's yield functions, of which the typically valid M is much higher than that of Hill's. FEM simulation of plane-strain sheet forming with 2008-T4 aluminium alloy verified that implementation of varying M-value with Hill's yield function led to better agreement with experimental measurements, while the variation of M with Barlat's yield function exhibited little influence on the strain prediction.     

Artificial aging and shear deformation behaviour of 6022 aluminium alloy

In this study, the artificial aging behaviour of 6022-T4 alloy is investigated over a wide temperature range. Hardness readings, TEM and XRD analyses were performed. It was shown that 6022-T4 alloy can be substantially hardened through a short aging treatment at temperatures in excess of 200 °C. The strain hardening curves of the 6022 alloy in different aging conditions were measured using the simple shear test and analysed in terms of their respective microstructures. The under-aged and pre-peak-aged exhibited a good combination of strength and strain hardening while the peak-aged alloy was characterised by maximum strength, albeit with a drastic reduction in strain hardening ability. Strain reversal experiments in simple shear were carried out in order to characterize the Bauschinger effect for the different heat treatment conditions. It was shown that the T4 and under-aged conditions lead to permanent softening of the flow stress.     

Development of viscoplastic constitutive equation through biaxial material testing

Internal-state variables have been used to represent the deformation historyin the recently proposed viscoplastic constitutive equations. In the current study, creep tests under nonproportional loadings were used to study the relative roles played by the internal-state varaibles in the constitutive equation by tracing the strain trajectory in strain space for a given stress trajectory in stress space. An experimental approach to studying the evolution rule for the tensorial state variable is also proposed. The experimental results on 2618-T61 aluminum alloy suggest that the scalarstate variable should play a much more dominant role than the tensorial state varaible in the constitutive modeling of 2618-T61 aluminum alloy.     

A simple model for dislocation behavior,strain and strain rate hardening evolution in deforming aluminum alloys

In this work, modeling of the stress–strain behavior is carried out using a simple dislocation model. This model uses three variables to characterize the dislocation population: The average forest and mobile dislocation densities, ρ_f and ρ_m, and the average dislocation mean free path L. However, it is shown that within reasonable assumptions, only two of these variables are independent. The mathematical form derived from this dislocation-based model was applied to experimental stress–strain data determined at room temperature for pure aluminum, 3003-O, 2008-T4, 6022-T4, 5182-O and 5032-T4 aluminum alloy sheets. The evolution of the state variables was calculated for these materials from a single stress–strain curve. The average dislocation mean free paths at a strain of 0.5 were compared with TEM observations of dislocation cell sizes or inter-dislocation spacing for specimens deformed equal biaxially with the hydraulic bulge test. A very good agreement was obtained between predictions and experiments.     

A Method of Modeling the Interaction of Creep and High-Cycle Fatigue

The interaction of creep and fatigue in structural materials under high-cycle loading is modeled using isochronic limit stress diagrams. The hypothesis of a unified limit diagram invariant to the time to failure is used. The unified diagram is given by a cosine power function with the exponent describing creep-fatigue interaction and encompasses convex, concave, and S-like curves. The models build are tested for aluminum alloys, heat-resistant steels, creep-resistant steels and alloys, and laminates__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 1, pp. 25–36, January 2005.     

Prediction of localized thinning in sheet metal using a general anisotropic yield criterion

The forming limit diagram (FLD) is a useful concept for characterizing the formability of sheet metal. The ability to accurately predict the FLD for a given material has been shown to depend on the shape of the selected yield function. In addition, both experimental and numerical results have shown that the level of the FLD is strongly strain path dependent. In this work, a combination of Marciniak–Kuczynski (M–K) analysis and a general anisotropic yield criterion developed by Karafillis and Boyce (Karafillis, A.P., Boyce, M.C., 1993. A general anisotropic yield criterion using bounds and transformation weighting tensor. J. Mech. Phys. Solids 41, 1859) is used to predict localized thinning of sheet metal alloys for linear and nonlinear strain paths. A new method for determining the constants in the yield criterion is proposed. The optimal values are obtained by fitting the initial yield stresses and calculated FLD under linear strain paths with the experimental measurement. Using this approach, accurate yield functions can be defined for both Al2008-T4 and Al6111-T4. Comparisons of computed FLDs with the experimental data of Graf and Hosford (Graf, A., Hosford, W.F., 1993b. Effect of changing strain paths on forming limit diagrams of Al 2008-T4. Metall. Trans. A. 24, 2503; Graf, A., Hosford, W.F., 1994. The influence of strain path changes on forming limit diagrams of Al 6111-T4. Int. J. Mech. Sci. 36, 897) show good agreements.     

Anisotropic strain hardening behavior in simple shear for cube textured aluminum alloy sheets

Finite element (FE) simulations of the simple shear test were conducted for 1050-O and 6022-T4 aluminum alloy sheet samples. Simulations were conducted with two different constitutive equations to account for plastic anisotropy: Either a recently proposed anisotropic yield function combined with an isotropic strain hardening law or a crystal plasticity model. The FE computed shear stress–shear strain curves were compared to the experimental curves measured for the two materials in previous works. Both phenomenological and polycrystal approaches led to results consistent with the experiments. These comparisons lead to a discussion concerning the assessment of anisotropic hardening in the simple shear test.     

Constitutive modeling of the material behavior at elevated temperatures and related material testing

Constant strain rate tests, both in tension and in torsion, were performed on 2618-T61 aluminum alloy at 200°C. The results were compared with those from the creep tests. It was demonstrated that the macroscopically observed material behavior was essentially a consequence of the inherent time dependence of the inelastic strain. Based on the experimental results, the traditional approach and the unified approach for modeling the material behavior at elevated temperatures were compared. The material tests required for the development of the constitutive model and their limitations were also discussed.     

Creep of a heat-resistant aluminum alloy in a complex state of stress

The article gives the results of experimental investigations of the creep of heat-resistant aluminum alloy AK4-1 with constant and variable loads, at a temperature of 175 °C and a duration of the tests equal to 100 h. Based on the experimental data, a verification of the theory of creep is adduced, based on the following hypotheses: 1) the change in the volume is elastic; 2) the deviator of the creep rates is proportional to the deviator of the stresses; 3) the intensities of the stresses, of the creep deformations, and their rates are connected by a relationship which does not depend on the type of the state of stress. It should be pointed out that the results of investigation of creep, under a complex state of stress, in carbon, low-alloy, austenitic steels, copper, and certain light alloys, are given in [1–6].Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 83–86, March–April, 1971.     

The fatigue of aluminum alloys subjected to random loading

This paper describes an expeirmental investigation which was carried out to determine the fatigue life of two aluminum alloys (2024-T3 and 6061-T6). They were subjected to both constant-strain-amplitude sinusoidal and narrow-band random-strain-amplitude fatigue loadings. The fatigue-life values obtained from the narrow-band random testing were compared with theoretical predictions based on Miner's linear accumulation of damage hypothesis. Cantilever-beam-test specimens fabricated from the aluminum alloys were subjected to either a constant-strain-amplitude sinusoidal or a narrow-band random base excitation by means of an electromagnetic vibrations exciter. It was found that the ε-N curves for both alloys could be approximated by three straight-line segments in the low-, intermediate- and high-cycle fatigue-life ranges. Miner's hypothesis was used to predict the narrow-band random fatigue lives of materials with this type of ε-N behavior. These fatigue-life predictions were found to consistently overestimate the acutal fatigue lives by a factor of 2 or 3. However, the shape of the predicted fatigue-life curves and the high-cycle fatigue behavior of both materials were found to be in good agreement with the experimental results.     

Alloying effects on dislocation substructure evolution of aluminum alloys

The constitutive response of aluminum alloys is controlled by the evolution of dislocation substructure including mobile and forest dislocation density, cell size distribution and morphology, and misorientation angle between neighboring cells. The present study focuses upon the small strain regime and compares the measured microstructural evolution of 3003, 5005, and 6022 aluminum alloys during deformation. Room temperature tensile deformation experiments were performed on industrially manufactured specimens of each alloy and the evolving microstructure was compared with the mechanical response. The dislocation structure evolution was characterized using transmission electron microscopy and orientation imaging of deformed specimens. It was observed that structural evolution is a function of lattice orientation and the character of neighboring grains. In general, the dislocation cell size and misorientation angle between dislocation cells evolves systematically with deformation at relatively small strain levels.     

In Situ Observation of Adiabatic Shear Band Formation in Aluminum Alloys

We used high-speed X-ray phase contrast imaging and infrared thermal imaging techniques to study the formation processes of adiabatic shear bands in aluminum 7075-T6 and 6061-T6 alloys. A modified compression Kolsky bar setup was used to apply the dynamic loading. A flat hat-shaped specimen design was adopted for generating the shear bands at the designated locations. Experimental results show that 7075-T6 exhibits less ductility and a narrower shear band than 6061-T6. Maximum temperatures of 720 K and 770 K were locally determined within the shear band zones for 7075-T6 and 6061-T6 respectively. This local high temperature zone and the resulting thermal instability were found to relate to the shear band formation in these aluminum alloys.