Evaluation of finite-plasticity theories for nonproportionate loading of torsion-tension members

Based on the assumption that the material was isotropic and even, and satisfied the condition of isotropic hardening for a von Mises material, finite-incremental and total-strain theories were derived for solid circularsection torsion-tension members subjected to nonproportionate loading. Torsion-tension members made of SAE 1045 steel and aluminum alloy 7075-T6 were subjected to proportionate and nonproportionate loading. During the nonproportionate loading, either the axial loal P or torque T was held constant while the other was increased. Excellent agreement was found between the incremental theory and experimental data indicating that the assumption of isotropic hardening is valid for this type of loading. For some of the nonproportionate loading paths, incremental and total-strain theories gave nearly identical results.     

Elevated-temperature creep and relaxation of torsion-tension members

Based on the assumption that each material satisfies the condition for isotropic hardening for a von Mises material, an incremental solution is developed to predict axial-strain creep curves and maximum shearing-strain creep and relaxation curves for solid circular torsion-tension members subjected to proportionate and nonproportionate stepped loading including creep in tension and relaxation in torsion. Test data are obtained from torsion-tension members made either of annealed OFHC copper at 800°F (427°C) or hot-rolled SAE 1045 steel at 950°F (510°C). The loading histories include either four stepped proportionate load changes, four stepped nonproportionate load changes, or torsion-tension loading in which the axial load remains constant and the torsional load is allowed to relax during two loading periods of the test. Each test duration is about 100 h. Good agreement is indicated between the predicted and measured creep and relaxation curves.     

Finite-strain pressure-tension loading of thick-walled cylinders including temperature effects

Experimental data were obtained from thick-walled cylinders made of hot-rolled SAE 1045 steel at room temperature, annealed OFHC copper at room temperature and at 500°F (260°C), and annealed aluminum alloy 1100 at room temperature and at 305° F (152° C). Experimental pressure-strain curves were compared with curves predicted by two different analytical solutions. One solution is a finite-strain, incremental, compressible analytical solution. The second solution is a finite-strain, total-strain, incompressible analytical solution which was corrected to make it applicable for compressible materials. With both solutions, the material is assumed to be an isotropic-hardening material that obeys the von Mises yield condition. The loading function for the material was obtained from tension specimens tested at the some temperature and loading rate as the thick-walled cylinders. Good agreement was found between each solution and experimental data.     

Creep of metal torsion-tension members subjected to nonproportionate load changes

The creep behavior of torsion-tension members subjected to nonproportionate loading was studied by testing members made of SAE 1035 steel at 975° F and of copper alloy 360 at 700° F. Creep curves for these members were predicted by an incremental theory using both the strain-hardening rule and the time-hardening rule and a total-strain theory using the incremental-hardening rule. Both incremental theories were based on a creep law which assumed that the total strain in specimens subjected to constant stress was made up of an elastic component and a creep component given by a power function of stress and of time. The total-strain theory was based on isochronous stress-strain curves which were approximated by an arc hyperbolic sine relation. The creep tests were limited to a total duration of 60 min. The nonproportionate loading was obtained by stepped increases of either the axial load, the torque, or both at 30 min. Except for the incremental theory based on the time-hardening rule, good agreement was found between theory and experiment.     

Unloading of thick-walled cylinders that have been plastically deformed

Two analytical solutions are used to predict load-strain relations for unloading of thick-walled cylinders. The solutions assume that the material is an isotropic-hardening material that obeys the von Mises yield condition. The loading function for the material for the unloading of the cylinders was obtained from tension-compression specimens that were unloaded and reverse loaded from several points along the tension stress-strain diagram. Good agreement is indicated between the unloading load-strain curves obtained from two thick-walled cylinders made of SAE 1045 steel and the curves predicted by the analytical solutions. The analytical solutions predict that the beneficial circumferential compressive residual stresses at the inside of the cylinders decrease by about 50 percent during the unloading.     

Experimental evaluation of incremental theories for nonproportionate loading of thin-walled cylinders

An experimental investigation was undertaken to evaluate incremental-strain theories which have been proposed in the literature to predict the loads on thin-walled cylinders subjected to nonproportionate loading which follow prescribed strain histories. Test data were obtained for two materials, annealed SAE 1035 steel and normalized 4340 steel. Material-property tests for the SAE 1035 steel indicated that the stress-strain diagram was flat topped and the material followed the Tresca flow condition. Similar tests for the SAE 4340 steel indicated that this steel was a linear strain-hardening material that followed the von Mises flow condition. Two incremental-strain theories were developed for thin-walled cylinders made of SAE 1035 steel. Both were based on the Tresca flow condition. One theory called the Tresca-Tresca theory used the stress-strain relations for the Tresca theory. The other theory called the Tresca-Mises theory used the Prandtl-Reuss stress-strain relations. In general, the test data fell between the two theories. The incremental theory developed for thin-walled cylinders made of the SAE 4340 steel, called the Mises-Mises theory, was based on the von Mises flow condition and the Prandtl-Reuss stress-strain relations. The agreement between theory and experiment was poor.     

Bursting pressure of thick-walled cylinders subjected to internal and external pressures,axial load and torsion

A finite-total-strain, incompressible, analytical solution is presented to predict load-deformation relations for loads from zero to failure for thick-walled cylinders subjected to internal pressure, external pressure, axial load and torsion. The solution assumes that the material is an isotropic hardening material that obeys the von Mises yield condition. The flow law incorporates the prandtl-Reuss stressstrain relations and a loading function represented by the tension true-stress vs. true-strain diagram. Poisson's ratio is assumed to be equal to one-half for both elastic and plastic strains. The difference between the strains given by the incompressible solution and the correct strains are calculated for one set of elastic loads; the strains given by the incompressible solution are then corrected based on the assumption that each correction is proportional to the increase in the given component of load. Good agreement is indicated between the corrected incompressible solution and data obtained from cylinders made of either SAE 1045 steel, OFHC copper, or aluminum alloy 1100. Paper was presented at 1975 SESA Spring Meeting held in Chicago, IL on May 11–16. This investigation was funded by Rock Island Arsenal and was conducted at Research Directorate, GEN Thomas J. Rodman Laboratory, under the Laboratory Research Cooperative Program of ARO-D.     

Incremental versus total-strain theories for proportionate and nonproportionate loading of torsion-tension members

Incremental and total-strain theories have been presented in the literature for hollow and solid circular torsion-tension members. The differential equations obtained for the incremental theory have been solved only for the conditions that the torsion-tension member is made of a nonstrain-hardening material and is subjected to restricted deformation histories. Computer programs were written to obtain numerical incremental solutions for hollow and solid circular torsion-tension members made of strain-hardening materials and subjected to any deformation or loading path. Test data were obtained for three different materials: (a) a nonstrain-hardening steel, annealed SAE 1035 steel, with identical properties in tension and compression; (b) a strain-hardening steel, annealed rail steel, with identical properties in tension and compression; (c) a strainhardening alumimum alloy, 2024-T4, with different properties in tension and compression. In all cases, the average of the tension and compression stress-strain diagram was approximated by two straight lines to obtain material properties. Test data for proportionate loading were in excellent agreement with either the total-strain theory or the incremental-strain theory. Data for nonproportionate loading, in which one deformation was kept constant as the other was increased, fell between the two theories and were in closer agreement with the predictions of the incremental theory.     

Strain-history effect on isotropic and anisotropic plastic behavior

An experimental investigation was performed to evaluate the effect of strain history on an initially isotropic material. A hot-rolled 2.5-in.-diam bar of SAE 1045 steel provided all the test specimens. Axial and circumferential compression data indicated that the steel was isotropic. Additional tension and torsion data indicated that the steel was an isotropic-hardening von Mises material; this was also confirmed by proportionate loading of thin-walled cylinders such that the ratio of axial to circumferential stresses was either 0, 1/2, 1, 2 or ∞. Two additional sets of cylinders were preloaded either in simple axial tension or as closed-ended cylinders to an effective plastic strain of 0.006 before they were proportionately loaded. The preloading had a pronounced effect on yield surfaces for reloading if the effective plastic strain on reloading was only slightly greater than that for the preloading. The effect of preloading on the yield surfaces was small when the effective plastic strain was three to four times that for the preloading. Hill's anisotropic theory was used to predict stress-strain relations for several of the reloaded cylinders. Good agreement was obtained between theory and experiment.     

Evaluation of finite-plasticity theories for torsion-tension members made of tresca materials

Finite-incremental Tresca and von Mises theories are developed for solid circular-section torsion-tension members subjected to proportionate and nonproportionate loading. The materials are assumed to be isotropic and even. Two Tresca theories and a von Mises theory are compared with test data obtained from torsion-tension members. Three different kinds of steels were tested; they are hot-rolled mild steel, annealed mild steel, and hot-rolled SAE 1017 steel. The fully plastic values of axial load and torque predicted by the Tresca theories agree with the experimental results; however, the deformations, in the strain-hardening region, predicted by both of the Tresca theories were greater than observed. The von Mises theory is nonconservative in predicting the fully plastic loads of torsion members and torsion-tension members and in predicting the deformations of torsion members in the strain-hardening region, but gives good correlation between predicted and experimental deformations for the torsion-tension members in the strain-hardening region.     

Photoelastic stress analysis of thick-walled closed-end cylinders

A three-dimensional photoelastic stress analysis was conducted on six thick-walled closed-end cylinders subjected to internal-pressure loading. The cylinders were approximately similar in design to the breech chamber of a large-caliber tank weapon. Maximum octahedral shear stresses were determined in the side wall-end wall juncture of essentially flat-ended cylinders (similar to actual weapon component) and in hemispherically-ended cylinders. The purpose of the study was to learn more about the stresses in the closed end of the cylinder so that future components of this type may be designed more efficiently with regard to stress and weight.     

Creep of thick-walled cylinders subjected to internal pressure and axial load

A creep theory is presented to predict deformations at any specified time for a thick-walled cylinder subjected to internal pressure and axial load. The theory is based on the usual assumptions that the deformations are infinitestimal, that the material is incompressible and that the total strain theory is valid. The stress-strain-time relation for the material is assumed to be represented by an isochronous stress-strain diagram which is approximated by an arc hyperbolic sine function. The experimental part of the investigation included tests of thick-walled cylinders made of high-density poly-ethylene whose ratio of outside to inside radii were either 1.5 or 2.0. The test cylinders were either tested as closed-ented cylinders with internal pressure or subjected to a combination of internal pressure and axial load. Also, the application of the theory for varying load conditions was studied. Good agreement was found between theory and experiment.     

A theory for swaging of discs and lugs

A practical theory for swaging bored holes within plates and cylinders is proposed which can take into account work-hardening in the presence of small plastic strains based upon equivalent stress-strain data. With the appropriate choice of yield function, this theory applies to the swaging of both thin and thick plates under respective plane stress and plane strain conditions. The theory can be adapted further to the autofrettage of open and closed-ended, thick-walled cylinders where similar plane deformations conditions apply. Here swaging refers to the practice in which an oversized plug or sphere is forced into the bore thereby expanding it permanently to leave a residual circumferential compression in the bore material upon removal of the expanding tool. A similar effect results from applying an initial over-pressure to a long thick-walled cylinder in an autofrettage process. Both treatments are employed to enhance the fatigue resistance when the service loading upon the disc or cylinder amounts to a cyclic, circumferential tension within its bore. Strain gauges bonded to the entry face of the plate are used to monitor the circumferential and radial strain distributions both during and after the swaging process. Experimental results presented for swaging of thin and thin annular discs in aluminium alloy show that the measured residual strain distributions concord with the theory for large discs with a 10/1 diameter ratio. The agreement is less satisfactory with the loss in axial symmetry for parallel-sided lugs with a width to hole diameter ratio of 4/1.     

Large deformation in polycrystalline copper under combined tension-torsion, loading, unloading and reloading or reverse loading: A study of two incremental theories of plasticity

Experimental data from combined tension-torsion of thin walled tubes of annealed polycrystalline copper subjected to various non-proportionate loading, unloading and reverse loading paths are presented. The measurements are compared with predicted values from classical incremental theory of plasticity in terms of true stress and true strain and a recently developed incremental theory of plasticity by Bell in terms of nominal stress and nominal strain. These experimental data reveal that the plastic strain produced by the various proportionate and non-proportionate loading, unloading and reverse loading paths are in better agreement with Bell's incremental theory of plasticity as compared to classical incremental theory.     

Instability and failure of internally pressurized ductile metal cylinders

Forlong, ductile, thick-walled tubes under internal pressure instabilities and final failure modes are studied experimentally and theoretically. The test specimens are closed-end cylinders made of an aluminum alloy and of pure copper and the experiments have been carried out for a number of different initial external radius to internal radius ratios. The experiments show necking on one side of the tubes at a stage somewhat beyond the maximum internal pressure. All tubes, except for one aluminum alloy tube, failed by shear fracture under decreasing pressure. The aluminum alloy tubes exhibited localized shear deformations in the neck region prior to fracture and also occasionally surface wave instabilities. The numerical investigation is based on an elastic-plastic material model for a solid that develops a vertex on the yield surface, using representations of the uniaxial stress-strain curves found experimentally. In contrast to the simplest flow theory of plasticity this material model predicts shear band instabilities at a realistic level of strain. A rather sharp vertex is used in the material model for the aluminum alloy, while a more blunt vertex is used to characterize copper. The theoretically predicted bifurcation into a necking mode, the cross-sectional shape of the neck, and finally the initiation and growth of shear bands from the highly strained internal surface in the neck region are in good agreement with the experimental observations.     

Large strain creep analysis of thick-walled cylinders

The finite-strain theory has been used to study the creep behaviour of a thick-walled cylinder under large strains. The analysis is divided into two parts. In part 1 the creep deformation of a thick-walled cylinder of an anisotropic material subjected to internal pressure has been discussed. The effect of the anisotropy has been depicted graphically. It is found that the anisotropy of the material has a significant effect on the axial stress, strain and strain rate. Part 2 of the paper deals with the creep analysis of cylinders of either isotropic or anisotropic materials subjected to combined internal and external pressures. The effect of the anisotropy is found to be similar to that found in part 1. It is seen, however, that the introduction of external pressure results in decreasing the strain rate and thus increasing the life of the cylinder.     

Edge impact of an elastic-plastic semi-infinite plate

The edge impact of an elastic-plastic semiinfinite plate subject to conditions of plane strain is investigated analytically and experimentally. The theoretical analysis is based on the strain-rate-independent theory of plastic-wave propagation. The plate is initially unstressed; the boundary condition for the edge of the plate corresponds to constant-velocity longitudinal impact except that the step in velocity has a finite rise time. Calculations are carried out according to both the approximate one-dimensional theory and the two-dimensional theory for an elastic-plastic isotropic work-hardening material. The rise time for both solutions is chosen so as to optimize the agreement between theoretical and experimental strain-time profiles. A numerical solution of the two-dimensional equations is obtained by using a difference method developed by Clifton;¹ the onedimensional approximation is solved by the well-known method of characteristics. The problem was approximated experimentally by the axial collision of 4-in.-diam annealed aluminum thick-walled cylinders with a diameter-to-wall-thickness ratio of ten. For impact velocities of 90, 130 and 160 ips (corresponding to maximum strains of 0.12, 0.22 and 0.36 percent, respectively), dispersive characteristics and maximum strain amplitudes of the strain wave are found to be in good agreement with the theoretical predictions of both solutions. However, the two-dimensional solution indicates that the stresses, strains and velocities in regions of high strain-rate are highly nonuniform across the plate thickness.     

A semi-analytic analysis of shape memory alloy thick-walled cylinders under internal pressure

In this paper, a new method for analysis of the pseudoelastic response of shape memory alloy thick-walled cylinders subjected to internal pressure is proposed. Two cases of short and long cylinders are considered by assuming the plane stress and plane strain conditions. In each case, a three-dimensional phenomenological SMA constitutive model is simplified to obtain the corresponding two-dimensional constitutive relations. The cylinder is partitioned into a finite number of narrow annular regions, and appropriate assumptions are made in order to find a closed-form solution for the equilibrium equations in each annular region. The global solution is obtained by enforcing the stress continuity condition at the interface of the annular regions and imposing the boundary conditions. Several numerical examples are presented to demonstrate the efficiency of the proposed method, and the results are compared with three-dimensional finite element simulations.     

The strain-rate effect and the incremental plastic wave in copper

An experimental investigation was conducted to determine the degree of sensitivity of commerically pure copper to strain rate and to note the effect of this sensitivity on the velocity of propagation of shearing strain in copper. Thin-walled cylindrical specimens of copper were loaded in torsion to eliminate the effects of radial inertia. All specimens were annealed and then cold worked in torsion to obtain necessary specimen uniformity. Quasi-static tests were performed on short-length specimens to determine the shearing stress-strain curve of copper at a very low strain rate. The strain-rate sensitivity of copper at low strain rates, from 3×10^?4/sec to 5/sec, was tested by loading short specimens at a very slow continuous rate and then suddenly increasing the strain rate. A quasi-static test was also performed to determine the effect of creep on prestressed copper. Dynamic tests involving strain rates up to 500/sec were performed on long specimens with a torsional impact machine. Specimens were tested under stress-free and prestressed initial conditions. The prestressed specimen was loaded at a slow, continuous rate before impact to avoid the undesirable effects of creep which would have occurred with a static preload. Results from the quasi-static tests showed that copper is noticeably sensitive to strain rate in the low strain-rate regions, but that the sensitivity becomes almost constant as the strain rate is increased. Results from the dynamic tests showed that large strains propagated at speeds which agreed well with speeds predicted by the strain-rate-independent theory of plastic-wave propagation. The lower-level strains in the prestressed specimen, however, propagated at much higher speeds than are predicted by the strain-rate independent. Because radial-inertia effects were not present, this discrepancy in measured and predicted speeds for low-level strains must be due to the strain-rate sensitivity of copper.     

Ratcheting of cyclically hardening and softening materials: II. Multiaxial behavior

Part II of this study is concerned with ratcheting phenomena of cyclically hardening and softening materials under biaxial, cyclic loading. Two sets of biaxial experiments were performed on carbon steel 1018 and stainless steel 304 thin-walled tubes. In the first tyoe of experiment, a constant internal pressure was prescribed while the tubes were cycled axially in a strain-symmetric fashion. This causes ratcheting in the circumferential direction. In the second type of experiment, the axial cycling was carried out under stress control. This loading history results in simultaneous ratcheting in the axial and circumferential directions. In the case of stainless steel 304, the nonproportionality of these loading histories was found to induce significant hardening in addition to that recorded in unaxial loading. Cyclic hardening was found to reduce the rate of ratcheting. In the case of carbon steel 1018, the nonproportionality of the loading paths was found not to influence the induced softening. Cyclic softening in the axial and circumferential directions were found to be uncoupled.The time-independent cyclic plasticity models developed in Part I, suitably extended to multiaxial loading, were used to simulate the biaxial ratcheting experiments. Two methods for modeling the additional hardening/softening of the material due to nonproportional loading, developed by previous investigators, were incorporated in the models. The prediction of circumferential ratcheting is shown again to be sensitive to the kinematic hardening rule of the yield surface incorporated in the models. The performance of the models in predicting the biaxial ratcheting results was found to be rather poor. Several reasons for this poor performance are identified and suggestions for future improvements are made.