Saturation Hardening Research Articles

Investigation of the influence of plane strain constrained anisotropic plastic flow on the localized necking prediction

PurposeThe purpose of this manuscript is to investigate the influence of the satisfying of plane strain conditions on the prediction of the localized necking and to evaluate this effect by the different hardening models.Design/methodology/approachHigh-order anisotropic Drucker yield function, two types of hardening models, namely Hollomon power and Voce saturated, and the modified maximum force criterion (MMFC) are employed to predict the forming limit strains of AA3104-H19 alloy. Two identification methods, namely conventional and plane strain constrained, are applied and forming limit diagrams of the material are predicted by the incorporation of the anisotropic Drucker criterion and the hardening models into the MMFC for both calibration methods.FindingsThe enhancement in the prediction accuracy of the forming limits provided by the implementation of the plane strain constrained method is strongly dependent upon the selected hardening model type in the MMFC. About 18% improvement in the prediction of the biaxial limit strains is provided via Hollomon power law hardening, whereas the opposite result is occurred in case that the Voce saturated hardening law is used, and the predictions match with the experimental data only in the plane strain point and its surroundings.Originality/valueThe plane strain constrained identification method precisely predicts the location of the plane strain point on the yield locus and also any yield stress or anisotropy coefficient in the plane strain state is not required during the calibration, therefore the method can reduce the number of the required experiments for the constitutive characterization, and it will be attractive for both academy and industry. Additionally, the improvement in the prediction of the biaxial forming limits can be provided by applying of this method along with the appropriate hardening law.

Experimental and modeling study of cyclic plasticity and ductile fracture of thin structural steel sheets

The cyclic material properties of thin steel sheets are critical for accurate prediction of the structural response of the thin-walled steel elements. For the experimental study of the cyclic behaviors of thin steel sheets, we proposed a new anti-buckling device equipped with the disc-springs. This improvement aims to compensate the potential gaps between the fixtures and specimens under large plastic deformation so as to avoid the out-of-plane buckling. Furthermore, the negative influences derived from the transverse pressure and friction can be greatly minimized through the implementation of the disc-springs and the lubricated Teflon sheet, as indicated by the percentage errors of 0.31% and 0.08% for the external load and internal stress respectively. By virtue of this improved device, we perform 8 regular and 1 random cyclic testing with the Q345 steel sheet of the 6 mm thickness. It is noted that the smooth hysteresis loops feature no decrease in force under compression applied. Moreover, the sizes of hysteresis loops are dependent upon the current and maximum strain amplitude. The current strain amplitude controls the re-yielding rate, while the maximum value determines the saturated hardening stress. As for the ductile fracture initiation, the similar nominal stress–strain curves in the final tension stage among all cases and the considerably larger accumulated plastic strain than the one under monotonic loading imply the contribution of negative triaxiality to the decrease in the damage variable. Based on these observations, we develop a new cyclic plasticity and ductile fracture model, which is also programmed into the UMAT subroutine compatible with ABAQUS/Standard. By the comparison with the experimental results, the remarkable agreement of the predicted results demonstrates its validity in terms of the constitutive relationship, deformed shape of necking region and the fracture initiation under regular and particularly random cyclic loading.

Characteristic stepwise strain hardening behaviour induced by slip and twinning of large-deformed copper single crystals: crystal plasticity modelling and simulation

ABSTRACT The micromechanism-based stepwise strain hardening behaviour of copper single crystals significantly depends on the competition and collaboration between dislocation slip and deformation twinning. A crystal plasticity-based model of the strain hardening characteristics associated with slip and twinning interactions is proposed that focuses on modelling the stepwise strain hardening behaviour. An accumulated twin volume fraction is incorporated into the evolution of slip resistance to model slip–twinning (S-T) interactions at small strains. Then, the following stress fluctuation stage caused by the crystal reorientation is related to the twinning–twinning (T-T) interactions. Upon deformation to a large strain, a saturated hardening law including the slip–slip (S-S) interactions in the twinned region and the accumulated slip are presented. The effects of the initial crystal orientation and twinning on distinguished hardening stages are investigated. The stepwise strain hardening model reveals the dominating hardening mechanism at the initial hardening stage, which has been identified for orientation-dependent copper single crystals by investigating the slip and twinning increments. The stress fluctuation of a copper single crystal with a particular grain orientation can be described by introducing the evolution of the twin volume fraction and its saturation value, which allows an in-depth understanding of the evolution of large plastic deformation.

Internal-variable constitutive model for rate-independent plasticity with hardening saturation surface

An elastic-plastic material model with internal variables and thermodynamic potential, not admitting hardening states out of a saturation surface, is presented. The existence of such a saturation surface in the internal variables space — a consequence of the boundedness of the energy that can be stored in the material's internal micro-structure — encompasses, in case of general kinematic/isotropic hardening, a one-parameter family of envelope surfaces in the stress space, which in turn is enveloped by a limit surface. In contrast to a multi-surface model, noad hoc rules are required to avoid the intersection between the yield and bounding/envelope surface. The flow laws of the proposed model are studied in case of associative plasticity with the aid of the maximum intrinsic dissipation theorem. It is shown that the material behaves like a standard one as long as its hardening state either is not saturated, or undergoes a desaturation from a saturated hardening state, whereas, for saturated hardening states not followed by desaturation, it conforms to a combined yielding law in which the static internal variable rates obey a nonlinear hardening rule similar to that of analogous models of the literature. Additionally, the material is shown to behave as a perfectly plastic material for a class of (critical) saturated hardening states for which the stress state is on the limit surface. For nonassociative material models, it is shown that, under a special choice of the plastic and saturation potentials and through a suitable parameter identification, the well-known Chaboche model is reproduced. A few numerical examples are presented to illustrate the associative material response under monotonic and cyclic loadings.

Elastic-Plastic Response of Sintered Porous Iron Under Uniaxial Strain Cycling

Elastic-plastic responses of porous iron under uniaxial strain cycling between two fixed values of strain are investigated. A special set of constitutive equations is formulated by including isotropic, kinematic and saturation hardening responses. The theoretical results from the constitutive equations are compared with experimental cyclic data for porous iron, with various porosities.

Structure properties of heavily cold-drawn niobium

The mechanical properties and microstructural evolution were evaluated for wires of niobium, cold drawn to true strains η up to 11.9. The ultimate tensile stress and fracture stress of niobium increases exponentially after drawing to η = 8.9. At larger true strains, up to η = 11.9 saturation hardening is observed in niobium as it is at much lower deformation strains ( η > 5.3) in copper. Niobium remains ductile after deformation processing to true strains up to η = 11.9, but at η ⩾ 10.3, failure occurs by ductile shear. Previous studies on heavily deformation-processed b.c.c. metals usually have been restricted to η<8.9, which is the most likely reason why saturation hardening was not previously observed in b.c.c. metals (J. Gil Sevillano et al., Progr. Mater., Sci., 25 (1981) 69; J. Gil Sevillano, Acta Metall., 34 (1986) 1473). Comparison of hardening in identically wire-drawn niobium and copper shows that the predominant difference between them was that copper exhibited saturation hardening after smaller draw ratios. Hardening in both metals appeared to be exponential in nature prior to the onset of saturation hardening. Analyses of the substructural development in niobium with increasing degree of deformation processing by wire drawing showed that grains and cells on transverse sections start with equiaxed shapes becoming ribbon-like and curled about the wire axis, tending toward a more equiaxed shape again with increasing deformation processing. The dislocation structures observed for niobium are typical of those observed in stage IV hardening and the substructural development with increasing deformation processing appears similar to previously reported observations on other b.c.c. metals.

A numerical simulation of cyclic hardening in copper.

The constitutive equation for analyzing the cyclic deformation was developed by introducing the concept of the internal stress and the dislocation multiplication into the Johnston-Gilman's equation modified by the authors. Then, the numerical simulation of the cyclic stress-strain hysteresis loop accompanied by the cyclic straining was done for the annealed copper. The stress and plastic strain ranges estimated were slightly different from the test results in the transition hardening stage under the total strain range-controlled condition. Meanwhile, the former was in good agreement with the latter in the saturation hardening stage, where the cyclic mechanical properties of materials were commonly represented in the practical use. The analyzed hysteresis loop tends to be fatter in shape than the experimental one at the larger strain range controlled in the test. This problem, however, is solved by adjusting one of the parameters included in the proposed constitutive equation, which are determined by the static tensile and stress-relaxation tests.

Compaction equations for strain hardening porous materials

Abstract Theoretical compaction equations for strain hardening porous materials are obtained by solving the problem of external pressurization of a hollow sphere of incompressible, rigid-plastic hardening material, for various forms of the hardening law. These hardening laws are chosen so that the integral expression for the pressure in the compaction problem can be evaluated in terms of elementary functions. This approach was adopted previously for a particular stress-strain response—the saturation hardening model of Voce and Palm—gave some theoretical compaction equations, including the empirical equation of Konopicky and Shapiro and Kolthoff. The present work treats nonsaturation models for which the ultimate strength is unbounded. The models presented here afford a much improved capability of fitting experimental strain hardening and compaction curves for harder materials. The theoretical predictions are compared with experimental data for sintered and unsintered porous beryllium, tungsten powder and alumina powder.

Two-Dimensional Strain Cycling in Plasticity

Detailed calculations are presented for strain cycling in a homogeneous deformation that can be sustained by a biaxial state of stress in thin-walled specimens of OFHC copper. These calculations are made with a set of relatively simple constitutive equations within the framework of the strain-space formulation of plasticity. The predicted theoretical calculations, carried out in the context of small deformation, are in good agreement with corresponding available experimental results for saturation hardening and erasure of memory in two-dimensional strain cycling. Also, with the use of the calculated results, a scalar quantity that characterizes strain-hardening is plotted as a function of plastic strains. Such plots are likely to be useful for computational purposes.

Calculations for Uniaxial Stress and Strain Cycling in Plasticity

Within the framework of an existing purely mechanical, rate-type theory of plasticity, detailed calculations are presented for certain types of material response during stress and strain cycling in a uniaxial homogeneous deformation. These features pertain specifically to material response in stress cycling between fixed values of stress in tension and compression (not necessarily equal in magnitude) resulting in ratcheting of strain, and a type of saturation hardening caused by strain cycling between any two fixed values of strain when the mean value of stress (in tension and compression) tends to zero.

On the Characterization of Strain-Hardening in Plasticity

In the context of a purely mechanical, rate-type theory of elastic-plastic materials and utilizing a strain space formulation introduced in [1], this paper is concerned mainly with developments pertaining to strain-hardening behavior consisting of three distinct types of material response, namely, hardening, softening, and perfectly plastic behavior. It is shown that such strain-hardening behavior may be characterized by a rate-independent quotient of quantities occurring in the loading criteria of strain space and the corresponding loading conditions of stress space. With the use of special constitutive equations, the predictive capability of the results obtained are illustrated for strain-hardening response and saturation hardening in a uniaxial tension test.

Effect of Grain Size on Low Cycle Fatigue in Low Carbon Steel

Low cycle fatigue tests were performed on a low carbon steel of 0.1 percent carbon content, which has various grain sizes of the range from 18 to l63μm in diameter. The cyclic softening is followed by a stationary state in cyclic deformation behavior in the fine grain material, while the coarse grain one exhibits a rapid cyclic hardening in the early stage, which is followed by a saturation hardening. Such a difference in cyclic deformation property can be interpreted well by considering that in the yielding behavior between the fine and the coarse grain materials. The cyclic stress-strain property is not influenced, by the grain size. The coarse grain material of 163μm in diameter has about 0.4times shorter fatigue life than that in the fine grain one of 18μm. The grain size dependent fatigue life of this kind is valid for both processes of crack initiation and propagation.

On the Hardening Response in Small Deformation of Metals

This paper is concerned with a special class of hardening response functions for small deformation of elastic-plastic materials, its application to isotropic metals, and comparison of the theoretical results with experimental cyclic stress-strain curves for two different metals. The theoretical development is carried out within the scope of the existing purely mechanical theory for the “rate-independent” response of elastic-plastic materials, which admits the existence of a single loading function, as well as certain accepted idealizations. After summarizing the basic equations for small deformation, detailed attention is given to the development of a special form of the hardening response function, motivated mainly by the observation that the stress-strain curves for uniaxial cyclic loading of a fairly large class of metals attain—after several cycles—the so-called saturation hardening. We exploit this property; and, in the case of isotropic metals, systematically derive some restrictions on the constitutive coefficients in the loading function and the hardening response. Comparison of the results with two sets of experimental data, obtained from uniaxial cyclic loading of a 304 stainless steel and a 2024 aluminum alloy, shows good agreement within the understood idealizations of the basic theory.

Mechanisms of fatigue hardening in copper single crystals

Copper single crystals were cycled in push-pull at a constant total shear-strain amplitude, γ t , of ±0.0075, which gave a fatigue life, N f , of 42,500 cycles. Dislocation arrangements were determined as a function of the number of elapsed cycles and were correlated with the fatigue-hardening curve. The cyclic stress-strain curve was determined over the range γ t = ±0.0016–0.016. Results have been compared with other recent fatigue-hardening studies. Dislocation bundles in dipole and multipole configurations are formed by a mutual-trapping mechanism during rapid hardening. The bundles contribute to rapid hardening by providing effective barriers to continued dislocation motion on the primary and secondary slip systems. Dislocation structures in the earliest stages of rapid hardening were found to be analogous to unidirectional dislocation structures; the similarity rapidly fades, however, with continued cycling and the progressive development of a cell structure, until at saturation hardening (≈0.5 per cent N f ), a dislocation structure uniquely characteristic of cyclic deformation is developed. Strain accommodation during saturation hardening can be accomplished by the cooperative movement of dislocations within cell walls (or loop patches) and by the traffic of dislocations across cells (or between loop patches), with ∼50 per cent of the dissipated energy estimated to be available for point-defect production. The present results lead to a unified view of fatigue hardening for high- and low-strain amplitudes. The dislocation structures in low-amplitude loop patches and in high-amplitude cell walls appear to be identical, each being a product of the early mutual-trapping mechanism. A correlation is suggested between the occurrence of the Bauschinger effect and the development of simple dislocation bundles (Stage I type dislocation structure). It is postulated that the Bauschinger effect is a result of the relaxation of dislocations within these dislocation bundles.