Subsequent Yield Surfaces Research Articles

Mechanical response of AZ31B magnesium alloy: Experimental characterization and material modeling considering proportional loading at room temperature

Tension and compression experiments have been performed to characterize the mechanical response of 1.57mm AZ31B-O sheet at room temperature. Five different sheet orientations were used to characterize the in-plane anisotropy under tensile loading conditions while cubic samples consisting of adhesively-bonded layers of sheet samples were used for compression testing along four sheet directions. During uniaxial tensile testing, the axial and transverse strain components were measured using two independent extensometers. A digital image correlation system was used to measure the strain components during compression testing. Both instantaneous and cumulative r-values were measured as they evolved with plastic strain. A strong, evolving asymmetry is observed. An evolving anisotropic/asymmetric continuum-based material model based on a Cazacu–Plunkett–Barlat (CPB)-type yield function is proposed to fit the material behavior as a continuous function of plastic strain. Considerable improvement in the representation of the material behavior is achieved as the number of stress transformations used in the CPB yield surface formulation is increased. To capture the evolution of the envelope of the subsequent yield surfaces, the anisotropy and asymmetry parameters are replaced with functions expressed in terms of plastic strain. The evolution parameters are found by minimizing the difference between the model predictions and the experiments at discrete plastic strain levels, using gradient search methods. A strain rate-independent elastic–plastic material model incorporating the evolving envelope of subsequent yield surface formulation has been developed and implemented within a commercial finite element package. The model reproduces the experiments initially used for fitting. The predictions of the developed material model are compared with the measured load–displacement and strain distributions from a three-point bending experiment. Improvement in the prediction of strain and forming forces is observed compared to the previously available non-evolving material models.

An Experimental Investigation and an Elastoplastic Constitutive Model for a Porous Rock

This paper presents a new constitutive model for the description of elastoplastic behaviour of a porous rock. In the first section, a brief summary of the experimental investigations is presented. The data obtained show that an important coupling exists between elastic and plastic deformation, and that the confining pressure has a great influence on the yielding process of the rock. In the second section, firstly, a plastic internal variable is defined, which is able to record the influence of the stress state on the progressive yielding process. Then, based on the plastic internal variable, the respective evolutions of the subsequent yield surface, dilatancy angle and elastic parameters are studied, and the applicability of the generalized orthogonal flow rule which connects the irreversible strain increment and the yield function are analyzed. The results show that the generalized orthogonal flow rule is not suitable for the porous rock during plastic deformation. Then, based on the experimental evidence, a general formulation of the model is presented. In the third section, simulations of the conventional triaxial compression tests are performed, and the results show that the proposed model is indeed capable of capturing the main features of the mechanical behaviour of the rock. Finally, the model is extended to simulate unloading confining pressure tests, and for which it gives reasonable results.

A Modified Storen-Rice Bifurcation Analysis of Sheet Metal Forming Limit Diagrams

Bifurcation analysis is a theoretical prediction approach to measure the FLD when the localized neck causes development of vertex on subsequent yield surface as was adopted by Storen-Rice. Some analyses lead to solutions for special cases such as zero and minimum extension. They offer an equation which needs to be optimized with respect to the minimum limit strain versus neck orientation for the whole domain of FLD. Moreover, the previous reported results for the left-hand side of FLD are not quite satisfactory. In this paper, a re-investigation into bifurcation analysis adopted by S-R lead to modified equations which significantly improved FLD and could be respected as a more general approach to find FLD theoretically. The derivation and optimization procedure of equations are indicated and discussed in detail. The predicted limit strains are studied for different work hardening coefficients and compared with Storen-Rice, Zhu and some experimental data and the obtained results show more agreement. Furthermore, the present restrictions and the required conditions for validation of the Zhu approach are fully discussed.

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.

A self-consistent approach describing the strain induced anisotropy: Case of yield surface evolution

The main goal of this work is to describe the evolution (expansion, translation, distortion and rotation) of the yield surface using a recent developed self-consistent model with small strains assumption. In fact, the ellipsoidal inclusion case in the simplified non-incremental interaction scheme has been developed by Abdul-Latif and Radi (2010) [4] showing the dependency of the elasto-inelastic behavior of polycrystals on the grain shape. The concept of yield surface is very important for evaluating the material work-hardening behavior beyond the yield point. Hence, a particular attention is made on the description of initial and subsequent yield surfaces. To show the predictive abilities of this model, these surfaces are simulated and then evaluated as a model response for different values of key model parameters (such as the grain aspect ratio α = a/ b where a and b are the half-axes of the grain and the viscous parameter γ) subjected to radial proportional loading. Then, the effect of the two parameters on these surfaces will be discussed. Subsequent yield surfaces showed different proportions of translation, expansion, distortion and rotation. Moreover, the influence of the pre-loading nature on the subsequent yield surface evolution and its rotation will be then examined under three different loading conditions of tension, torsion, and combined tension–torsion proportional loading path. Some granular responses for pre-selected grains are also illustrated and discussed.

A finite deformation model for evolving flow anisotropy with distortional hardening including experimental validation

The paper presents a new constitutive model for anisotropic metal plasticity that takes into account the expansion or contraction (isotropic hardening), translation (kinematic hardening) and change of shape (distortional hardening) of the yield surface. The experimentally observed region of high curvature (“nose”) on the yield surface in the loading direction and flattened shape in the reverse loading direction is modeled here by means of the concept of directional distortional hardening. The modeling of directional distortional hardening is accomplished by means of an evolving fourth-order tensor. The applicability of the model is illustrated by fitting experimental subsequent yield surfaces at finite plastic deformation. The simulation results were compared with test data for aluminum low and high work hardening alloys. In this introductory paper we will consider only the proportional loading case.

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.

Description of yield surface evolution using a convected plasticity model

The convected plasticity model proposed by Wu (2003a,b, 2005, 2007), making use of convected coordinate system, is applied to discuss the evolution of yield surface. It is shown that this constitutive model is capable of describing all experimentally observed features of subsequent yield surface: isotropic hardening, kinematic hardening, distortion, and rotation of yield surface. The rotation of yield surface in 3D stress space has not been much discussed in the literature, but recent experiments at National Taiwan University ( Sung et al., 2011) have shown that it is an important property of subsequent yield surface. In particular, the rotation of subsequent yield surface is pre-strain path dependent. It does not rotate, when the pre-strain is tensile; but the yield surface rotates about the axial stress axis when the pre-strain is torsional.

Constitutive modelling of fibre-reinforced composites with unidirectional plies using a plasticity-based approach

This paper presents the development of a constitutive model able to accurately represent the full non-linear mechanical response of polymer–matrix fibre-reinforced composites with unidirectional (UD) plies under quasi-static loading. This is achieved by utilising an elasto-plastic modelling framework. The model captures key features that are often neglected in constitutive modelling of UD composites, such as the effect of hydrostatic pressure on both the elastic and non-elastic material response, the effect of multiaxial loading and dependence of the yield stress on the applied pressure. The constitutive model includes a novel yield function which accurately represents the yielding of the matrix within a unidirectional fibre-reinforced composite by removing the dependence on the stress in the fibre direction. A non-associative flow rule is used to capture the pressure sensitivity of the material. The experimentally observed translation of subsequent yield surfaces is modelled using a non-linear kinematic hardening rule. Furthermore, evolution laws are proposed for the non-linear hardening that relate to the applied hydrostatic pressure. Multiaxial test data is used to show that the model is able to predict the non-linear response under complex loading combinations, given only the experimental response from two uniaxial tests.

An elasto-plastic damage constitutive theory and its prediction of evolution of subsequent yield surfaces and elastic constants

Based on pair functional potentials, Cauchy–Born rule and slip mechanism, a material model assembling with spring-bundle components, a cubage component and slip components is established to describe the elasto-plastic damage constitutive relation under finite deformation. The expansion/shrink, translation and distortion of yield surfaces can be calculated based on the hardening rule and Bauschinger effect defined on the slip component level. Both kinematic and isotropic hardening are included. Numerical simulations and predictions under tension, torsion, and combined tension–torsion proportional/non-proportional loading are performed to obtain the evolution of subsequent yield surfaces and elastic constants and compare with two sets of experimental data in literature, one for a very low work hardening aluminum alloy Al 6061-T6511, and another for a very high work hardening aluminum alloy annealed 1100 Al. The feature of the yield surface in shape change, which presents a sharp front accompanied by a blunt rear under proportional loading, is described by the latent hardening and Bauschinger effect of slip components. Further, the evolution law of subsequent yield surfaces under different proportional loading paths is investigated in terms of their equivalence. The numerical simulations under non-proportional loading conditions for annealed 1100 Al are performed, and the subsequent yield surfaces exhibit mixed cross effect because the kinematic hardening and isotropic hardening follow different evolution tendency when loading path changes. The results of non-proportional loading demonstrate that the present model has the ability to address the issue of complex loading due to the introduction of state variables on slip components. Moreover, as an elasto-plastic damage constitutive model, the present model can also reflect the variation of elastic constants through damage defined on the spring-bundle components.

Evolution of yield surface in the 2D and 3D stress spaces

Initial and subsequent yield surfaces for 6061 aluminum, determined by a method of automated yield stress probing, are presented in the 2D ( σ zz − σ θz ) and 3D ( σ θθ − σ zz − σ θz ) stress spaces. In the ( σ zz − σ θz ) space, yield surfaces at small pre-strains show the noses and unapparent cross effect. At larger pre-strains, they become ellipses with positive cross effect. In the ( σ θθ − σ zz − σ θz ) space, the initial yield surface is not well described by von Mises yield criterion due to material anisotropy. The yield surfaces of various torsional pre-strains show obvious rotation around the σ zz axis but they do not rotate when subjected to axial pre-strains. Therefore, the rotation behavior of yield surface is pre-strain path dependent. The rotation of yield surfaces in the 3D space is the emphasis of the present paper. Coupled axial–torsional behavior subjected to torsion after axial pre-strain are also presented for the same material that is used to determine the yield surfaces. This information is useful for verification of constitutive models.

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.

Theoretical construction of initial and subsequent yield surfaces for isotropic strain-hardening elastoplastic materials of the differential type

Within the theory of infinitesimal strains, a new approach is proposed for the theoretical construction of the initial and subsequent loading surfaces for simple in Noll’s sense initially solid strain-hardening materials of the differential type of complexity 1. The loading surfaces are constructed without making additional assumptions. For the whole class of materials considered, the initial surface has the Huber–Mises form, the subsequent surface undergoes expansion, translation and distortion in the general case. In special cases, the equation of the subsequent loading surface corresponds to the equations of the well-known theories of plasticity with the isotropic-kinematic and kinematic hardening. It is established that the associated flow rule is valid for the initial loading surfaces and regular points of the subsequent loading surfaces. With the accepted definitions of the active loading, unloading and neutral loading, the Drucker postulate is valid for the initial surface and subsequent surfaces of smooth convex shape. The experiments required for the particular definition of the derived dependences are verified.

Microstructural effects on yield surface evolution in cubic metals using the viscoplastic ϕ-model

Polycrystalline yield surfaces of metals are a good way to characterize the anisotropy of plastic deformation. The evolution of these surfaces is impossible to accurately reproduce without taking into account the evolution of the material microstructure such as texture development. In this paper, a numerical computation of yield surfaces using the viscoplastic ϕ-model is proposed. Results concerning face-centered cubic metals subjected to a plane strain compression test are presented. The influence of several mechanical parameters (strain hardening, strain rate sensitivity coefficient and accumulated deformation) on subsequent yield surfaces evolution is studied. The analysis of the change in the shape and size of the yield surfaces shows that the results depend strongly on the parameter ϕ which controls the strength of the interactions in the polycrystal. In addition, the predictions are compared to the widely used viscoplastic self-consistent model as well as to experimental yield loci taken from the literature for various aluminum alloy sheets. A fairly good qualitative agreement between our ϕ-model results and the experimental ones is found. The probable links between the parameter ϕ and the microstructural features such as the stacking fault energy and the grain size of the polycrystal are also briefly discussed.

Experimental investigation of initial and subsequent yield surfaces for laminated metal matrix composites

The aim of this work is to construct yield surfaces to describe initial yielding and characterize hardening behavior of a highly anisotropic material. A methodology for constructing yield surfaces for isotropic materials using axial–torsion loading is extended to highly anisotropic materials. The technique uses a sensitive definition of yielding based on permanent strain rather than offset strain, and enables multiple yield points and multiple yield surfaces to be conducted on a single specimen. A target value of 20 × 10 −6 is used for Al 2O 3 fiber reinforced aluminum laminates having a fiber volume fraction of 0.55. Sixteen radial probes are used to define the yield locus in the axial–shear stress plane. Initial yield surfaces for [0 4], [90 4], and [0/90] 2 fibrous aluminum laminates are well described by ellipses in the axial–shear stress plane having aspect ratios of 10, 2.5, and 3.3, respectively. For reference, the aspect ratio of the Mises ellipse for an isotropic material is 1.73. Initial yield surfaces do not have a tension–compression asymmetry. Four overload profiles (plus, ex, hourglass, and zee) are applied to characterize hardening of a [0/90] 2 laminate by constructing 30 subsequent yield surfaces. Parameters to describe the center and axes of an ellipse are regressed to the yield points. The results clearly indicate that kinematic hardening dominates so that material state evolution can be described by tracking the center of the yield locus. For a nonproportional overload of ( σ, τ) = (500, 70) MPa, the center of the yield locus translated to ( σ, τ) = (430, 37) MPa and the ellipse major axis was only 110 MPa.

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.

Polycrystalline behavior analysis of pure magnesium by the homogenization method

In this study, mechanical behaviors of pure magnesium polycrystals are numerically investigated. The homogenization method, which combines the crystal and macroscopic scales, is introduced to include the effect of crystalline scale behaviors. The polycrystal plasticity model modified for pure magnesium, in which twinning is considered as asymmetric slip-like deformation, is utilized as a constitutive equation. Within this framework, numerical convergence analyses are conducted, and a representative volume element to present realistic deformation of pure magnesium is investigated. Second, polycrystalline behaviors of pure magnesium are investigated. The present approach is shown to reproduce the typical phenomena induced by crystalline scale structure in pure magnesium: nonuniform strain distribution, asymmetric crystal lattice orientation, strength differential effect, and strongly anisotropic initial and subsequent yield surfaces.

Evolution of subsequent yield surfaces and elastic constants with finite plastic deformation. Part III: Yield surface in tension–tension stress space (Al 6061–T 6511 and annealed 1100 Al)

Subsequent yield surfaces for aluminum alloys are determined for three proportional loading paths (i.e., axial, hoop, and combined hoop and axial stress) using 10 με deviation from linearity as the definition of yield. This paper is in continuation with Parts I and II of the author’s previous papers on subsequent yield surfaces under tension–torsion ( σ11–√3 σ 12) stress space using similar small offset definition of yield. In the current paper comprehensive experimental results on subsequent yield surfaces under tension–tension ( σ 11– σ 22) stress space are presented. Comparison of subsequent yield surfaces under ( σ 11–√3 σ 12) stress space, investigated in the earlier papers, clearly indicated distinctive hardening behavior under various loading paths. However, subsequent yield surfaces for Al 6061–T 6511 (a low work hardening alloy) showed contraction and negative cross-effect with finite deformation as compared to the annealed 1100 Al (a high work hardening alloy) where expansion and positive cross-effect was observed.

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.

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, A.S., Pandey, A., Stoughton, T., 2009a. Evolution of subsequent yield surfaces and elastic constants with finite plastic deformation, Part-II: a very high work hardening aluminum alloy (annealed 1100 Al). Int. J. Plast., accepted for publication; Khan, A.S., Pandey, A., Stoughton, T., 2009b. Evolution of subsequent yield surfaces and elastic moduli under tension–tension stress space with finite plastic deformation in aluminum alloys, Part-III. Int. J. Plast., accepted for Publication] 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.