Evolution of subsequent yield surfaces and elastic constants with finite plastic deformation. Part-I: A very low work hardening aluminum alloy (Al6061-T6511)

In the present study, the initial and subsequent yield surfaces in Al 6061-T6511, based on 10 με deviation from linearity definition of yield, are presented. The subsequent yield surfaces are determined during tension, free end torsion, and combined tension–torsion proportional loading paths after reaching different levels of strains. The yield surfaces are also obtained after linear, bi-linear and non-linear unloading paths after finite plastic deformation. The initial yield surface is very close to the von-Mises yield surface and the subsequent yield surfaces undergo translation and distortion. In the case of this low work hardening material, the size of the yield surfaces is smaller and negative cross-effect is observed with finite plastic deformation. The subsequent yield have a usual “nose” in the loading direction and flattened shape in the reverse loading direction; the observed nose is more dominant in the case of tension and combined tension–torsion loading than in torsional loading. The size of the yield surfaces after unloading is smaller than the initial yield surface but larger than subsequent yield surfaces obtained during prior loading, show much smaller cross-effect, and the shape of these yield surfaces depends strongly on the loading and unloading paths. Elastic constants (Young’s and shear moduli) are also measured within each subsequent yield surfaces. Evolution of these constants with finite deformation is also presented. The decrease of the two moduli is found to be much smaller than reported earlier in tension by Cleveland and Ghosh [Cleveland, R.M., Ghosh, A.K., 2002. Inelastic effects on springback in metals. Int. J. Plast. 18, 769–785]. Part-II and III [(Khan et al., 2009a) and (Khan et al., 2009b)] of the papers will include experimental results on annealed 1100 Al (a very high work hardening material) and on both Al alloys (Al6061-T6511 and annealed 1100 Al) in tension- tension stress space, respectively. The results for both cases are quite different than the ones that are presented in this paper.     

Evolution of subsequent yield surfaces and elastic constants with finite plastic deformation. Part II: A very high work hardening aluminum alloy (annealed 1100 Al)

Results are presented on the evolution of subsequent yield surfaces with finite deformation in a very high work hardening annealed 1100 aluminum alloy. In Part I [Khan, A.S., Kazmi, R., Stoughton, T., Pandey, A., 2009a. Evolution of subsequent yield surfaces and elastic constants with finite plastic deformation. Part 1: a very low work hardening aluminum alloy (Al-6061–T6511) 25, 1611–1625.] of this paper, similar results are presented for a very low work hardening aluminum alloy. Those results were very different from the present ones, and all the results were for proportional loading paths. The subsequent yield surfaces are determined in tension, free end torsion and combined tension–torsion proportional and non-proportional loading paths, using 10 με deviation from linearity definition of yield. Yield surfaces are also determined after linear, bi-linear, and non-linear unloading paths after finite deformation under tension, free end torsion, and combined tension–torsion loading. The initial yield surface is closer to the von-Mises surface and the subsequent yield surfaces show distortion, expansion, positive cross-effect, and “nose” in the loading direction. Additionally, the subsequent yield surfaces after non-proportional loading paths show shrinkage and compounded distortion. The yield surfaces after unloading depict strong anisotropy, positive cross-effect and exhibits different proportion of distortion in each loading conditions. The Young’s and shear modulus decrease with plastic deformation and this decrease is much less than those reported in the published literature.     

An experimental study on subsequent yield surface after finite shear prestraining

An equimodulus surface is introduced and the subsequent yield surface after large finite shear prestraining is experimentally investogated. Fully annealed, thin-walled copper tubular specimens were subjected to large torsional loading and partial unloading; strain gages were carefully mounted on the specimen after the application of pure shear loading. Specimens were then subjected to various combined tension-torsion loadings. Influences of he von Mises and Tresca equivalent offset strains on the subsequent yield surfaces are studied. On examining the experimental results reported in this article, it was found that the smaller the offset strains, the more distorted are the subsequent yield surfaces. At the torsional preloading point, a rounded corner was developed, whereas in the region opposite to the preloading point, the subsequent yield surface was flattened. When large von Mises offset strains were used, the corresponding subsequent yield surfaces passed through the von Mises loading surface. But this was not the case when Tresca offset strains were used. The subsequent yield surface determined by the back extrapolation method was almost completely outside the von Mises loading surface. On the other hand, the subsequent yield surface determined by the back extrapolation method was close to the Tresca loading surface. It is also found that the equimodulus surface is distorted and cannot simply be described by the combined kinematic and isotropic hardening rule.     

An endochronic theory accounting for deformation induced anisotropy of metals under biaxial load

Using the experimental results of yield surfaces obtained by Wu and Yeh [1991] (Int. J. Plasticity, 7, 803) for 304 stainless steel, this work provides a verification of the endochronic theory of plasticity accounting for deformation induced anisotropy. The experiments were performed under proportional loading conditions. The main difference between this paper and other papers that attempt to describe the distortion of a yield surface is that, in addition to distortion, motion of yield surface (kinematic hardening) has also been addressed by this paper. The result has shown that the theory predicts the experimental data with substantial accuracy. However, since in this theory the plastic strain increment, although normal to the initial yield surface, is in the radial direction emanating from the center of the subsequent yield surface, validity of the present model must be further studied for the case involving nonproportional loading conditions.     

Plastic deformation modes in perforated sheets and their relation to yield and limit surfaces

Construction of mechanism-based plasticity theories for the homogenized response of heterogeneous materials requires identification of plastic deformation modes as a function of loading direction relative to the microstructural details. Herein, we employ an efficient homogenization theory to identify for the first time such deformation modes in plates under plane stress with hexagonal arrays of circular holes at small and intermmediate pore volume fractions, and establish their relation to the branches of initial and subsequent yield and limit surfaces. Newly introduced maps of the intrinsic geometric features of point-wise yield surfaces provide full-field picture of the investigated microstructures’ propensity for plastic strain initiation and localization. The identified characteristic plastic modes provide a rational explanation for the evolving geometric features of subsequent yield and limit surfaces whose branches represent different plastic flow mechanisms, as well as a basis for the construction of a mechanism-based homogenized plasticity theory for use in structural analysis algorithms. The results suggest the need for composite yield surfaces comprised of multiple branches in the construction of mechanism-based homogenized plasticity theory for the investigated class of porous materials.     

Research note on a reexamination of neutral loading experiments

In the examination of the published results from neutral loading experiments, the question as to whether plastic deformation occurs is found to depend on both the material and initial loading strain. Provided that initial loading is elastic, then a subsequent stress path that follows the boundary of the initial yield surface for a hardening material is truly neutral with a wholly elastic response. However, when initial loading is elastic-plastic, then further plastic deformation is produced from a subsequent stress path that follows an isotropic expansion of the initial yield surface. These results enable the appropriateness of the kinematic hardening rule and more recent developments in plasticity theory to be appraised. Neutral loading of a non-hardening material produces plastic flow. Whether the absence of hardening is inherent or induced by plastic prestrain, it is shown that the Prandtl-Reuss theory then represents the observed behavior. In general, the purely elastic and nonhardening solutions provide respectively lower and upper bounds on the deformation.     

Experimental investigation of the behavior of extra high strength steel

This paper comprises a study concerning the mechanical behavior of extra high strength steel. This is investigated by means of biaxial testing of flat cross-shaped specimens in the full σ₁-σ₂ plane, a concept developed earlier at Steel Structures, Luleå University of Technology. Furthermore, new specimen designs had to be developed to enable testing of a material with high yield strength and low ultimate over yield strength ratio, such as the extra high strength steel Weldox 1100. The tests are performed in two steps: one initial loading followed by unloading and a subsequent loading in a new direction. The test results, containing data from 15 biaxial tests, are characterized by a slightly anisotropic initial yield criterion where the proof stress in compression is consequently somewhat higher compared to the results in tension. In the subsequent loading the observed phenomena are a Bauschinger effect in the direction opposite the initial loading direction and that the transition from elastic to plastic state in subsequent loadings is gradual and direction-dependent.     

A more comprehensive method for yield locus construction for metallic alloys and composites

Conventional methods for constructing yield loci rely on the assumption that nonlinear strains are permanent strains, which is not always the case. A nickel-base alloy, SiC fiber-reinforced titanium, an aluminum alloy, and particlereinforced aluminum have been observed to violate this assumption. We present a method for constructing yield loci using a proof strain criterion for the permanent strain that relies on cyclic, proportional, probes of the yield surface. Two criteria are implemented: one for stress reversal and one for yielding. The method is demonstrated by the construction of initial and subsequent yield loci in the axial-shear stress plane using thin-walled tubular specimens. Results are presented for 6061-T6 aluminum as well as for 6092/SiC/17.5p-T6, which is 6092 aluminum reinforced with 17.5 volume percent silicon carbide particulate. The centers of the initial yield loci for the composite are eccentric to the origin of the stress plane most likely because of the residual stresses induced during processing. Material hardening due to multiaxial stress states can be described by tracking evolution of the subsequent yield surfaces and here hardening of the particulate composite was primarily kinematic     

YIELD SURFACES AND PLASTIC FLOW OF 45 STEEL UNDER TENSION-TORSION LOADING PATHS

An experimental analysis on the subsequent yield-surfaces evolution using multiple specimens is presented for a 45 steel after a prescribed pre-strain loading in three different directions respectively, and the yielding is defined by a designated offsetting strain. The size of the subsequent yield surface is found smaller than the initial yield surface; the negative cross effects are observed in the normal loading direction, its shape is not a Mises circle but has a rather blunt nose in loading direction and flat in the opposite. These results strongly depend on the loading path and the prescribed offset plastic strain. The plastic flow direction to the subsequent yield surface is investigated, and it is found that the plastic flow direction deviates from the normal flow rule. The deviation differs from preloading case to preloading case. And the plastic flow direction would have a larger deviation from the normal of the yield surface, if the subsequent yield was defined by a smaller offset strain. Furthermore, the experiments are simulated using the Chaboche model, and the results show that it can rationally predict yield-surface only when yield is defined by a fairly large offset strain.     

On the experimental determination of yield surfaces and some results of annealed 304 stainless steel

Factors affecting the experimental determination of yield surfaces are discussed. They include the elastic moduli and the zero offset strain, the strain domain used to determine the yield stress, the probing path, and the strain rate of probing. To obtain yield surfaces consistently, it is necessary to account for these factors. The initial and subsequent yield surfaces of annealed AISI type 304 stainless steel have been experimentally determined in the axial-torsional stress space. Three loading paths have been studied. They are a pure axial path, a pure torsional path, and a proportional axial-torsional path. Each path includes loading, unloading, reloading, and the cyclically steady state.     

Localization in elasto-plastic materials: Influence of the plasticity yield surface in biaxial loading conditions

The influence of the plasticity yield surface on the development of instabilities in plane plates in biaxial loading is analyzed in order to understand and simulate the localization pattern observed in an expanding hemisphere experiment. First, a criterion for the activation of slip bands is formulated in the form of a critical hardening coefficient: it is particularized to the Von Mises and Tresca surfaces. In the Von Mises case, the criterion gives a strongly negative hardening coefficient in biaxial loading conditions different from the ones of plane strain. In the Tresca case, the criterion is fulfilled for a perfectly plastic material in uniaxial and biaxial loading; besides, in equi-biaxial loading, two possible orientations for slip bands are exhibited; this can be understood, with a few approximations, by the existence of a vertex point on the Tresca yield surface which give additive degrees of freedom for the direction of the plastic strain rate. Second, the development of localization in the loading conditions met in an expanding hemisphere experiment is simulated using both plasticity yield surfaces; whereas the Von Mises simulation does not localize, the Tresca simulation exhibits a pattern composed of a network of shear bands of different orientations; this pattern is not far from the pattern observed experimentally.     

A plasticity model for pressure-dependent anisotropic cellular solids

The initial and subsequent yield surfaces for an anisotropic and pressure-dependent 2D stochastic cellular material, which represents solid foams, are investigated under biaxial loading using finite element analysis. Scalar measures of stress and strain, namely characteristic stress and characteristic strain, are used to describe the constitutive response of cellular material along various stress paths. The coupling between loading path and strain hardening is then investigated in characteristic stress–strain domain. The nature of the flow rule that best describes the plastic flow of cellular solid is also investigated. An incremental plasticity framework is proposed to describe the pressure-dependent plastic flow of 2D stochastic cellular solids. The proposed plasticity framework adopts the anisotropic and pressure-dependent yield function recently introduced by Alkhader and Vural [Alkhader M., Vural M., 2009a. An energy-based anisotropic yield criterion for cellular solids and validation by biaxial FE simulations. J. Mech. Phys. Solids 57(5), 871–890]. It has been shown that the proposed yield function can be simply calibrated using elastic constants and flow stresses under uniaixal loading. Comparison of stress fields predicted by continuum plasticity model to the ones obtained from FE analysis shows good agreement for the range of loading paths and strains investigated.     

A general cyclic plasticity model taking into account yield surface distortion for multiaxial ratchetting

Most of the theoretical studies devoted to multiaxial ratchetting are focused on the location of the macroscopic yield domain in order to predict the correct direction of plastic strain rate, since normality of the plastic strain rate to the yield surface is usually assumed. To the authors' knowledge, the shape of subsequent yield surfaces was always kept constant in these models for simplicity reasons but unlike experimental observations. In a previous paper [J. Eng. Mater. Technol. 124(4) (2002) 402], the authors have explained the need to take into account yield surface distortion in macroscopic modeling and have therefore proposed such a constitutive model but only for biaxial loadings. In the present paper, the generalization of this distortional model is proposed for any loading paths. The model is then identified and validated on a large data base obtained firstly with an efficient polycrystalline model that can predict multiaxial ratchetting as well as yield surface distortion, and secondly with the experimental results of complex tests realized on a type 316L stainless steel.     

A study on subsequent yield surface based on the distributed-element model

This study considers extension of the distributed-element model to account for the deformation-induced anisotropy demonstrated in subsequent yield surfaces. By modifying the shape parameter of the strength-distribution function in the model as a function of the accumulated plastic deformation and the preloading direction, the DEM is able to account appropriately for major features of the subsequent yield surfaces demonstrated by real materials under proportional loading. The proposed modeling technique is confirmed by performing numerical simulations and comparing with experimental results available in the literature.     

Yield surfaces of various periodic metal honeycombs at intermediate relative density

Initial yield surfaces are derived for several periodic metal honeycomb cell structures with sufficiently high relative density that failure occurs by plastic yielding. Both in-plane stress states (normal stresses perpendicular to cell axes, with in-plane shear) and triaxial stress states with one principal stress direction along the cell axes are considered. Beam/column and plate/shell yield criteria are adopted to address general in-plane loading and 3D triaxial loading, respectively, accounting for combined cell wall stretching and bending as appropriate. Cell wall behavior is assumed to be elastic-perfectly plastic. The initial yield surfaces for different periodic cell structures are systematically compared. Some issues related to the initial yield surfaces of various honeycombs are discussed, including dependencies on relative density and in-plane and out-of-plane applied stresses, as well as the influence of joints between cell walls.     

Size-effects on yield surfaces for micro reinforced composites

Size effects in heterogeneous materials are studied using a rate independent higher order strain gradient plasticity theory, where strain gradient effects are incorporated in the free energy of the material. Numerical studies are carried out using a finite element method, where the components of the plastic strain tensor appear as free variables in addition to the displacement variables. Non-conventional boundary conditions are applied at material interfaces to model a constraint on plastic flow due to dislocation blocking. Unit cell calculations are carried out under generalized plane strain conditions. The homogenized response of a material with cylindrical reinforcing fibers is analyzed for different values of the internal material length scale and homogenized yield surfaces are presented. While the main focus is on initial yield surfaces, subsequent yield surfaces are also presented. The center of the yield surface is tracked under uniaxial loading both in the transverse and longitudinal directions and an anisotropic Bauschinger effect is shown to depend on the size of the fibers. Results are compared to conventional predictions, and size-effects on the kinematic hardening are accentuated.     

A Ring-Shaped Plastic Zone at an Elliptic Crack under Triaxial Axisymmetric Loading

The paper addresses a three-dimensional problem for an elliptic crack with a ring plastic zone under uni- and triaxial loading at infinity. The normal stresses in the plastic zone are found from the conditions that the stresses are constrained and the plane strain is local and from the yield criterion for the given material. The size of the elliptic ring is calculated by Rice's variational formula. It is shown that the constraint ratio for plastic strains under triaxial loading may be greater than that under loads close to hydrostatic tension. The contour of the plastic zone is confocal to the initial crack