Showing papers on "Strain hardening exponent published in 1987"

Dislocation-mechanics-based constitutive relations for material dynamics calculations

Abstract: An improved description of copper‐ and iron‐cylinder impact (Taylor) test results has been obtained through the use of dislocation‐mechanics‐based constitutive relations in the Lagrangian material dynamics computer program EPIC‐2. The effects of strain hardening, strain‐rate hardening, and thermal softening based on thermal activation analysis have been incorporated into a reasonably accurate constitutive relation for copper. The relation has a relatively simple expression and should be applicable to a wide range of fcc materials. The effect of grain size is included. A relation for iron is also presented. It also has a simple expression and is applicable to other bcc materials but is presently incomplete, since the important effect of deformation twinning in bcc materials is not included. A possible method of acounting for twinning is discussed and will be reported on more fully in future work. A main point made here is that each material structure type (fcc, bcc, hcp) will have its own constitutive beha...

Measurement of the temperature profile during shear band formation in steels deforming at high strain rates.

Abstract: A torsional Kolsky bar (split-Hopkinson bar) was used to deform tubular specimens of AISI 1018 cold rolled steel and AISI 1020 hot rolled steel at a nominal strain rate of 103s−1. Shear bands were observed to form in both steels, and the temperature of the material in the bands was measured by determining the infrared radiation emitted at the metal surface. For this purpose, a linear array of ten indium-antimonide detectors was used to determine temperature history at ten neighboring points lying across the projected path of the shear band. Results showed that shear bands in these steels are relatively wide, that the maximum temperature rise in the band is about 450°C and that the temperature distribution across the band is consistent with results of stability analyses. The two steels have very different work hardening rates and the strain at which localization is first observed is very different for the two steels : in the cold-rolled steel it occurs at about 15% strain, while in the hot-rolled the strain is near 100%. This result also is consistent with predictions of the analyses.

Analytical Characterization of Shear Localization in Thermoviscoplastic Materials

Abstract: : Critical conditions for shear localization in thermoviscoplastic materials are obtained in closed form for idealized models of simple shearing deformations. The idealizations, which include the neglect of heat conduction, inertia, and elasticity, are viewed as quite acceptable for many applications in which shear bands occur. Explicit results obtained for the idealized, but fully nonlinear problem show the roles of strain rate sensitivity, strain hardening, and initial imperfection on the localization behavior. Numerical solutions for two steels are shown to exhibit the principal features reported for torsional Kolsky bar experiments on these steels. Mathematically exact critical conditions obtained for the fully nonlinear problem are compared with critical conditions obtained by means of linear perturbation analysis gives better agreement with the predictions fo the fully nonlinear analysis.

Onset of shear localization in viscoplastic solids

Abstract: An outstanding problem in mechanics is the modeling of the phenomenon of initiation and development of localized shear bands, in materials whose inelastic deformation behavior is inherently rate-dependent. C lifton (1980) and B ai (1981, 1982) have presented a linear perturbation stability analysis for the initiation of shear bands in viscoplastic solids deforming in simple shear. This localization analysis is essentially one-dimensional in nature. In this paper, we present a three-dimensional generalization of this linear perturbation stability analysis for the onset of shear localization. We neglect elastic effects and consider isotropic, incompressible, viscoplastic materials which exhibit strain hardening (or softening), strain-rate hardening, thermal softening and pressure hardening. For this class of materials, we derive the general characteristic stability equation. Next, we focus our attention on the special version of this equation for plane motions and consider the two physically important limiting cases of (1) quasi-static, isothermal deformations, and (2) dynamic, adiabatic deformations. For these cases, we derive (a) the critical conditions for the formation of shear bands, (b) the most probable directions along which the bands can form, and (c) information regarding the incipient rate of growth of the emergent shear bands. The results, which are presented in detail in the body of the paper, provide new insight into the important phenomenon of shear localization under both quasi-static and rapid deformation of many materials including both metals and polymers.

An Evaluation of Recent Developments in Hardening and Flow Rules for Rate-Independent, Nonproportional Cyclic Plasticity

Abstract: The Mroz kinematic hardening rule has previously demonstrated superior capability to correlate cyclically stable nonproportional stress-strain response. In this paper, recently proposed kinematic hardening rules for single and multiple surface cyclic plasticity models are evaluated. Significant improvement over the Mroz rule, without loss of generality, is achieved with a deviatoric stress rate-dominated rule proposed by Tseng and Lee for two surface theory. Recent approaches for correlation of the modulus function and isotropic hardening are discussed. The norm of the Mroz distance vector is found to uniquely correlate the variation of plastic hardening modulus through a cycle; it is necessary to include a measure of instantaneous nonproportionality, however, to properly normalize the modulus function. A new evolution equation is offered to correlate the additional isotropic hardening observed during nonproportional loading, and several contemporary approaches are also considered.

Mechanical behavior of aluminum-lithium alloys at cryogenic temperatures

Abstract: The cryogenic mechanical properties of aluminum-lithium alloys are of interest because these alloys are attractive candidate materials for cryogenic tankage. Previous work indicates that the strength-toughness relationship for alloy 2090-T81 (Al-2.7Cu-2.2Li-0.12Zr by weight) improves significantly as temperature decreases. The subject of this investigation is the mechanism of this improvement. Deformation behavior was studied since the fracture morphology did not change with temperature. Tensile failures in 2090-T81 and -T4 occur at plastic instability. In contrast, in the binary aluminum-lithium alloy studied here they occur well before plastic instability. For all three materials, the strain hardening rate in the longitudinal direction increases as temperature decreases. This increase is associated with an improvement in tensile elongation at low temperatures. In alloy 2090-T4, these results correlate with a decrease in planar slip at low temperatures. The improved toughness at low temperatures is believed to be due to increased stable deformation prior to fracture.

Asymptotic Fields in Steady Crack Growth with Linear Strain-Hardening,

Abstract: The asymptotic stress and velocity fields of a crack propagating steadily and quasi-statically into an elastic-plastic material are presented. The material is characterized by J 2 -flow theory with linear strain- hardening. The possibility of reloading on the crack flanks is taken into account. The cases of anti-plane strain (mode III), plane strain (modes I and II), and plane stress (modes I and II) are considered. Numerical results are given for the strength of the singularity and for the distribution of the stress and velocity fields in the plastic loading, elastic unloading and plastic reloading regions, as functions of the strain-hardening parameter. An attempt is made to make a connection with the perfectly-plastic solutions in the limit of vanishing strain-hardening.

Ductile two-phase alloys: Prediction of strengthening at high strains

Abstract: When alloys containing two ductile phases are heavily deformed, composite-like microstructures develop and strengths well in excess of either of the phases in single phase form may be exhibited as a result of microstructure/dislocation density effects. In this paper a previously-published model for such strengthening is reviewed, and its application in a predictive capacity discussed. Flow stressvs fabrication strain data for the two components in single phase form and for one two-phase alloy are necessary for this purpose. The model may then be applied to predict strength for any other two-phase alloy as a function of composition, fabrication strain, and interphase spacing. The approach is illustrated using existing data for several alloy systems. For Ag-Fe and Cu-Nb alloys (with very limited mutual solubility) strengths can be predicted within 15 to 20 pct of the experimental values over the entire range of strains and volume fractions for which data are available. In systems where the potential for precipitation hardening exists (e.g., Cu-Fe) thermal history is important. When such hardening becomes a significant factor, the model cannot be used in its present form due to uncertainty over how to “add” the strengthening from this effect. Such hardening may, however, be useful in further improving the properties of these materials.

A Physically Based Internal Variable Model for Rate Dependent Plasticity

Abstract: In 1978 Krieg et al. published a ‘unified creep-plasticity’ model for rate-dependent deformation of metals, incorporating a power-law relationship between inelastic strain rate, applied stress, and instantaneous value of two internal variables.1 The internal variables were permitted to evolve by a Bailey—Orowan process, including strain hardening and recovery. Hardening was taken to be linear and to increase the internal variables rather than the flow stress directly; recovery was treated as thermal (as opposed to dynamic strain-activated) only, where the kinetics were derived from dislocation mechanics for the process in question. The physical basis was established because (a) the power-law flow rule was taken to be a mathematically convenient approximation to rate-process theory at fixed microstructural state, (b) linear strain hardening in polycrystals is usually viewed as an aggregate manifestation of the linear (stage II) hardening behavior of fcc single crystals oriented initially for single slip and in the absence of dynamic recovery,2,3 and (c) the recovery kinetics were derived from dislocation models. The value of physical bases follows, of course, from the confidence (indeed the meaning) that is given to extrapolation of the relationship beyond the range of measured data.

Polymer melt elongation—methods, results, and recent developments

Abstract: For film blowing of polyethylene it has been shown previously that melt elongation is very powerful for polymer characterization. With two types of rheometers, simple (also called “uniaxial”) elongational tests as well as creep tests can be performed homogeneously. In simple elongation, the melts of branched polyethylene show a remarkable strain hardening. With respect to their advantages and disadvantages, these rheometers complement each other. For multiaxial elongations the various modes of deformation can be performed by means of the rotary clamp technique. With the strain rate components ordered such that 11 ⩾ 22 ≥ 33, the ratio m = 22/11 characterizes the test mode. The Stephenson definition of the elongational viscosities makes use of the linear viscoelastic material equation and proves to be very efficient because the linear shear viscosity (t) (“stressing” viscosity) can act as the reference for the nonlinear behavior in elongation. Results are given for polyisobutylene measured not only in simple, equibiaxial, and planar elongations, but also in new test modes with a change of m during the deformation. This allows one to investigate the consequences of a deformation-induced anisotropy of the rheological behavior.

Overload retardation in a structural steel

Abstract: The mechanisms causing crack growth retardation after an overload were examined for BS4360 50B steel. It was found that plasticity-induced crack closure is the main cause of retardation when the pre-overload growth rate is in the mid-regime of the growth rate versus stress intensity range plot. When the pre-overload growth rate is near threshold it is argued that retardation at the surface of the specimen is primarily due to strain hardening and to the build-up of a favourable residual stress distribution in the material ahead of the crack tip. Supporting evidence for this argument is provided by a preliminary test on 2014A-T4 aluminium alloy. Plasticity-induced crack closure may be a further cause of retardation in the bulk, plane strain regions of the specimens made from BS4360 50B steel and 2014A-T4 aluminium alloy, when the pre-overload growth rate is near threshold.

Void/Pore Distributions and Ductile Fracture.

Abstract: The dependence of ductile, microvoid fracture on the size and distribution of voids or pores has been modeled experimentally. Pores or voids have been physically modeled in two dimensions by both random and regular arrays of equi-sized holes drilled through the thickness of tensile specimens of 1100-0 Al sheet and 7075-T6 Al plate and sheet. Fracture strains as well as failure paths have been determined for different hole sizes, spacings, and area fractions. A statistical analysis of the data indicates that increasing the minimum hole spacing, which decreases the degree of hole clustering, increases both strength and ductility. Conversely, decreasing the hole size causes a minor increase in both strength and ductility. Increasing the rate of work hardening is beneficial to ductility in that a high strain hardening rate appears to increase the resistance to flow localization between holes. The results are discussed in terms of a fracture process which depends on shear localization between holes/voids and which is very sensitive to void/pore distributions.

Linear low density polyethylenes and their blends: Part 3. Extensional flow of LLDPE's

Abstract: The uniaxial extensional flow at 150°C of 11 linear low density polyethylenes (LLDPE) and one low density polyethylene was measured in a Rheometrics Extensional Rheometer. The presence of silicone oil did not affect the results. However, large effects of the molding time were observed. For specimens molded for 14 min, strain hardening was not observed for any gas-phase polymerized LLDPE. As the molding time was increased to 40 min, the strain hardening was quite apparent, the elongational viscosity nearly doubled, the equilibrium plateau vanished, and the maximum strain at break Increased by about 20 percent. Explanation for the molding time effects can be found in the concept of low entanglement density in the virgin gas-phase resins. The entanglement increases with time at temperatures above the melting point. The specimens molded for longer time show strain hardening.

Cyclic hardening mechanisms in NIMONIC 80A

Abstract: A nickel base superalloy was fatigued under constant plastic strain range (Δep) control. The hardening response was investigated as a function of Δep and particle size of the γ ′ phase. Hardening was found to be a function of the slip band spacing,i. Numerous measurements ofi and other statistical data on the slip band structures were obtained. Interactions between intersecting slip systems were shown to influence hardening. A Petch-Hall model was found to describe best this relationship between the response stress and the slip band spacing.

Comparison of the rheology of polymer melts in shear, and biaxial and uniaxial extensions

Abstract: The experimental properties of different polymer melts, polystyrene, high density polyethylene and low density polyethylene are compared for the first time in three different deformations: step shear, step biaxial extension and steady uniaxial extension. Properties of three other melts are also studied in step biaxial and shear experiments. For our comparative purposes some data of Laun and Winter from the literature are used, as well as new data reported here. In all the step strain experiments, the stresses can be factored into a time dependent relaxation modulus and a strain dependent damping function. The data are interpreted using a differential constitutive equation of Larson which satisfies this time-strain separability and has a single parameter that describes the strain softening character of the material. Results show that differences in the properties of the melts are most pronounced in uniaxial extension and least in biaxial extension. All melts follow the Doi-Edwards prediction relatively closely in biaxial extension. In uniaxial extension, the branched material shows a strong strain hardening effect although its shear and biaxial properties are similar to the other melts. The constitutive model gives a reasonably good fit to the data in all three deformations for unbranched materials for the same value of the adjustable parameter; the model, however, fails for the branched low density polyethylene.

Theory of mobile dislocation density: Application to the deformation of 304 stainless steel

Abstract: Tensile and creep deformation of 304 stainless steel have been studied at room temperature in a soft tensile machine. The strain-stress curve shows a long inelastic transient, over nearly two hundred megapascals, in which the slope and the strain rate increase by about a factor of 3. Creep is characterized by large creep strain, transient strain rate, and a weak stress dependence. The stress rate applied during loading, on the other hand, has a strong effect on the subsequent creep. These results, in combination with earlier studies, have suggested a new model for the evolution of mobile dislocation density. Mobile dislocations are injected into the sample as a consequence of the increase of stress; the dislocations move under the influence of an effective stress over a statistical mean free path and then are trapped in a network. The effects of trapping are reduced mobile density, strain hardening, and a decrease in the effective stress. Application of the model provides a quantitative prediction of the principal experimental results.

A simplified model of heat generation during the uniaxial tensile test

Abstract: The temperature rise in a sheet tensile specimen has been calculated by the finite difference method for a plain-carbon steel at various strain rates and in several environments. Prior to necking, a uniform heat generation function is used with the governing flow equation while during the post-uniform strain, an empirical heat generation function is used. The empirical function is based on a strain distribution equation generated by curve fitting of experimental data. The effect of heat transfer conditions on the temperature increase has been discussed. The maximum temperature rise in air may reach 42 K at the center of an I.F. steel specimen at a strain rate of 10-2/s. The instability strain during tensile testing has been predicted by taking into account strain hardening, strain-rate hardening, and deformationinduced heating. The results show that significant deformation heating can occur during tensile testing in air at “normal” strain rates near 10-2/s, and that the uniform elongation can be affected markedly. Predictions for other alloys based on tabulated data are also presented.

Theory of plastic and viscous deformation

Abstract: A theory of inelastic deformation, previously applied to 304 stainless steel with good quantitative agreement,2 is used to study a variety of materials which differ in the degree to which dislocation motion is resisted by viscous drag forces. In the theory, mobile dislocations are injected into the material by the rising stress, move over a mean free path to create strain, and are trapped. The velocity of motion, determined by the magnitude of an effective stress relative to the viscous drag, determines the mean lifetime of mobile dislocations and thereby, in part, the mobile density. An attractive feature of the theory is its simplicity. There are only three significant physical constants, two which characterize the dislocation velocity and one, taken in this case to be material independent, which determines the strain-hardening coefficient. The calculations have been done to simulate a variety of tests done in a soft tensile machine, in which the principal control is exerted over the rate of stress increase. The results show diversetransient strain rate behavior, determined by the magnitude of the drag forces, but a commonsteady state strain rate, controlled by strain hardening. Soft materials with low viscous drag, such as copper, exhibit brief transients on change of stress rate, whereas in hard materials with high drag, such as iron-3.5 pct silicon, the transients are very long. These transients include the onset of yielding at the start of a strain-stress test, low temperature creep, and the strain rate response to a brief pulse of high stress rate. Thus for example, hard materials show long loading transients (slow approach to steady state), extensive low temperature creep, and no evident ‘rapid’ strain during a high rate stress pulse. For soft materials the converse results obtain. These differences and others distinguish, respectively, viscous and plastic deformation behavior.

A study in the work-hardening behaviour of austenitic manganese steels

Abstract: The work-hardening behaviour of Hadfield's steel and the effect of vanadium additions have been examined. At room temperature, deformation twinning plays a significant role. The rate of work-hardening was found to be sensitive to the presence of vanadium above 1 wt%. Transmission electron microscopy studies indicate that the work-hardening is due to a combination of structural defects. The mechanical behaviour of the alloys in compression can be described by a mathematical expression derived from the familiar Ludwik expression.

On the contribution of concurrent grain growth to strain sensitive flow of a superplastic Al-Cu eutectic alloy

Abstract: Flow behavior and microstructural evolution in an Al-Cu eutectic alloy of equiaxed grains were investigated over e ≃ 2× 10−6 to 2 × 10−2 s−1 andT = 400° to 540 °C Depending on the test conditions, there appeared either strain hardening or strain softening predominantly in the early part of the σ-e curves The microstructural observations showed evidence for grain growth, development of zig-zag boundaries, dislocation interactions, and cavitation The grain growth adequately accounts for the observed strain hardening at higher temperatures and lower strain rates However, at lower temperatures the strain hardening can be only partly accounted for by the observed grain growth; under this condition, some dislocation interactions also contribute to the strain hardening The presence of cavitation causes strain softening predominantly at higher strain rates Therefore, to develop a proper understanding of the superplastic behavior of the Al-Cu eutectic alloy, it is necessary to take into account the influence of dislocation interactions and cavitation along with that of grain growth

Constitutive equations for rigid plastic or viscoplastic materials containing voids

Abstract: — A systematic numerical study is reported that uses a homogenization technique to consider the plastic and viscoplastic potential of a porous medium; the study also takes account of the interactions between cavities. Calculations are carried out in plane strain with cylindrical cavities for various stress states characterized by the ratio of the principal stresses and for various values of the porosity as well as of the strain hardening exponent of the matrix. It is shown that the cavity growth rate is a strong function of the porosity, of the strain hardening exponent of the matrix and that it is a linear function of the stress triaxiality ratio. It is demonstrated that the growth models in an infinite body of Rice and Tracey or of Budiansky et al. can underestimate the cavity growth rate for intermediate stress triaxiality ratios, even for porosities as low as 1%.