Analyzing the orientation dependence of stresses in polycrystals using vertices of the single crystal yield surface and crystallographic fibers of orientation space

The influence of the polygonal geometry of the restricted slip-associated yield surface on the distribution of stresses over a polycrystalline aggregate is examined. The vertices of the yield surface (stress states corresponding to polyslip) are grouped according to symmetries imposed by crystal structure. A measure of coaxiality between crystal stresses and yield surface vertex stresses is used to quantify the proximity of the stress in each crystal to a yield surface vertex. It is shown that for prescribed stress states, crystal stresses align more closely with certain families of vertices than with others and this relation between crystal and vertex stresses is found to depend on crystallographic fibers. Using this information, the stress distributions from finite element simulations of face centered cubic polycrystals are analyzed for different stress states ranging from uniaxial to balanced biaxial. Over the fundamental region of orientations, the propensity for the stress to align with a vertex is demonstrated. Further, the stresses in elements contributing to certain crystallographic fibers are shown to favor the vertex families aligned with those fibers. The implications of these results on mechanical behaviors, especially with respect to those observed in diffraction experiments, are discussed.     

A finite element formulation based on non-associated plasticity for sheet metal forming

In the present paper, a finite element formulation based on non-associated plasticity is developed. In the constitutive formulation, isotropic hardening is assumed and an evolution equation for the hardening parameter consistent with the principle of plastic work equivalence is introduced. The yield function and plastic potential function are considered as two different functions with functional form as the yield function of Hill [Hill, R., 1948. Theory of yielding and plastic flow of anisotropic metals. Proc. Roy. Soc. A 193, 281–297] or Karafillis–Boyce associated model [Karafillis, A.P. Boyce, M., 1993. A general anisotropic yield criterion using bounds and a transformation weighting tensor. J. Mech. Phys. Solids 41, 1859–1886]. Algorithmic formulations of constitutive models that utilize associated or non-associated flow rule coupled with Hill or Karafillis–Boyce stress functions are derived by application of implicit return mapping procedure. Capabilities in predicting planar anisotropy of the Hill and Karafillis–Boyce stress functions are investigated considering material data of Al2008-T4 and Al2090-T3 sheet samples. The accuracy of the derived stress integration procedures is investigated by calculating iso-error maps.     

Necking in glassy polymers: Effects of intrinsic anisotropy and structural evolution kinetics in their viscoplastic flow

Two recently proposed developments of the Glass–Rubber constitutive model for glassy polymers treat the viscoplastic deformation as intrinsically anisotropic, and incorporate the kinetics of structural evolution. These features enable the model to capture better the distinctive features of glassy polymers’ constitutive response: post-yield strain-softening and strain-hardening and effects of pre-existing molecular orientation. They have been combined to form a new variant of the model, and the consequences for necking have been explored. Uniaxial extension of prismatic bars was simulated using the finite element method, employing a numerical implementation of the new model, with material parameters of polystyrene. Strain localization predicted with the new model was found to be systematically retarded as compared to predictions with the original (intrinsically isotropic) version of the model, for the same conditions. In particular, the effect of frozen-in molecular orientation was examined. This was found to retard strain localization for stretching parallel to the orientation direction, for both models. But the localization predicted with the new model was always significantly less pronounced than with the original model. Indeed, for sufficiently high pre-orientation (e.g. a uniaxial stretch of 2.2), localization could be effectively prevented with the new model, under conditions when otherwise failure by necking is predicted. Such results can all be explained in terms of a linear stability analysis. They suggest that all previous simulations of necking in glassy polymers made using intrinsically isotropic representations of polymer viscoplasticity may have over-predicted the rate of strain localization.     

Multiscale modeling of plasticity based on embedding the viscoplastic self-consistent formulation in implicit finite elements

This paper is concerned with the multiscale simulation of plastic deformation of metallic specimens using physically-based models that take into account their polycrystalline microstructure and the directionality of deformation mechanisms acting at single-crystal level. A polycrystal model based on self-consistent homogenization of single-crystal viscoplastic behavior is used to provide a texture-sensitive constitutive response of each material point, within a boundary problem solved with finite elements (FE) at the macroscale. The resulting constitutive behavior is that of an elasto-viscoplastic material, implemented in the implicit FE code ABAQUS. The widely-used viscoplastic selfconsistent (VPSC) formulation for polycrystal deformation has been implemented inside a user-defined material (UMAT) subroutine, providing the relationship between stress and plastic strain-rate response. Each integration point of the FE model is considered as a polycrystal with a given initial texture that evolves with deformation. The viscoplastic compliance tensor computed internally in the polycrystal model is in turn used for the minimization of a suitable-designed residual, as well as in the construction of the elasto-viscoplastic tangent stiffness matrix required by the implicit FE scheme.Uniaxial tension and simple shear of an FCC polycrystal have been used to benchmark the accuracy of the proposed implicit scheme and the correct treatment of rotations for prediction of texture evolution. In addition, two applications are presented to illustrate the potential of the multiscale strategy: a simulation of rolling of an FCC plate, in which the model predicts the development of different textures through the thickness of the plate; and the deformation under 4-point bending of textured HCP bars, in which the model captures the dimensional changes associated with different orientations of the dominant texture component with respect to the bending plane.     

A three-dimensional crystal plasticity model for duplex Ti–6Al–4V

A rate dependent crystal plasticity model for the α/β Ti–Al alloy Ti–6Al–4V with duplex microstructure is developed and presented herein. Duplex Ti–6Al–4V is a dual-phase alloy consisting of an hcp structured matrix primary α-phase and secondary lamellar α + β domains that are composed of alternating layers of secondary α laths and bcc structured residual β laths. The model accounts for distinct three-dimensional slip geometry for each phase, anisotropic and length scale dependent slip system strengths, the non-planar dislocation core structure of prismatic screw dislocations in the primary α-phase, and crystallographic texture. The model is implemented in the general purpose finite element code (ABAQUS, 2005. Ver 6.5, Hibbitt, Karlsson, and Sorensen, Inc., Pawtucket, RI) via a UMAT subroutine.     

Crystal-plasticity finite-element analysis of inelastic behavior during unloading in a magnesium alloy sheet

A crystal-plasticity finite-element analysis of the loading-unloading process under uniaxial tension of a rolled magnesium alloy sheet was carried out, and the mechanism of the inelastic response during unloading was examined, focusing on the effects of basal and nonbasal slip systems. The prismatic and basal slip systems were mainly activated during loading, but the activation of the prismatic slip systems was more dominant. Thus the overall stress level during loading was determined primarily by the prismatic slip systems. The prismatic slip systems were hardly activated during unloading because the stress level was of course lower than that during loading. On the other hand, because the strength of the basal slip systems was much lower than that of the prismatic slip systems, the basal slip systems would be easily activated under the stress level during unloading in the opposite direction when their Schmid’s resolved shear stresses changed signs because of the inhomogeneity of the material. These results indicated that one explanation for the inelastic behavior during unloading was that the basal slip systems were primarily activated owing to their low strengths compared to that of the prismatic slip systems. Numerical tests using the sheets with random orientations and with the more pronounced texture were conducted to further examine the mechanism.     

Numerical simulations of necking during tensile deformation of aluminum single crystals

Finite element simulations are used to study strain localization during uniaxial tensile straining of a single crystal with properties representative of pure Al. The crystal is modeled using a constitutive equation incorporating self- and latent-hardening. The simulations are used to investigate the influence of the initial orientation of the loading axis relative to the crystal, as well as the hardening and strain rate sensitivity of the crystal on the strain to localization. We find that (i) the specimen fails by diffuse necking for strain rate exponents m < 100, and a sharp neck for m > 100. (ii) The strain to localization is a decreasing function of m for m < 100, and is relatively insensitive to m for m > 100. (iii) The strain to localization is a minimum when the tensile axis is close to (but not exactly parallel to) a high symmetry direction such as [1 0 0] or [1 1 1] and the variation of the strain to localization with orientation is highly sensitive to the strain rate exponent and latent-hardening behavior of the crystal. This behavior can be explained in terms of changes in the active slip systems as the initial orientation of the crystal is varied.     

Crystal plasticity modeling of slip activity in Ti–6Al–4V under high cycle fatigue loading

Deformation micromechanisms of a Ti–6Al–4V alloy under fatigue loading at room temperature are studied using a three-dimensional crystal plasticity constitutive model. The model employs a minimum set of fitting parameters based on experimental data for Ti–6Al–4V. Single slip is strongly favored through a softening law that affects mainly the driving force for slip on the first activated slip system. Cyclic deformation behavior at the macroscopic scale and at the local scale of grains is analyzed through the simulation of 20 cycles of fatigue on a polycrystalline structure of 900 randomly oriented grains. The progressive activation of slip (basal, prismatic, and pyramidal) is analyzed and compared to experimental observations.     

A non-associated constitutive model with mixed iso-kinematic hardening for finite element simulation of sheet metal forming

In this paper an anisotropic material model based on non-associated flow rule and mixed isotropic–kinematic hardening was developed and implemented into a user-defined material (UMAT) subroutine for the commercial finite element code ABAQUS. Both yield function and plastic potential were defined in the form of Hill’s [Hill, R., 1948. A theory of the yielding and plastic flow of anisotropic metals. Proc. R. Soc. Lond. A 193, 281–297] quadratic anisotropic function, where the coefficients for the yield function were determined from the yield stresses in different material orientations, and those of the plastic potential were determined from the r-values in different directions. Isotropic hardening follows a nonlinear behavior, generally in the power law form for most grades of steel and the exponential law form for aluminum alloys. Also, a kinematic hardening law was implemented to account for cyclic loading effects. The evolution of the backstress tensor was modeled based on the nonlinear kinematic hardening theory (Armstrong–Frederick formulation). Computational plasticity equations were then formulated by using a return-mapping algorithm to integrate the stress over each time increment. Either explicit or implicit time integration schemes can be used for this model. Finally, the implemented material model was utilized to simulate two sheet metal forming processes: the cup drawing of AA2090-T3, and the springback of the channel drawing of two sheet materials (DP600 and AA6022-T43). Experimental cyclic shear tests were carried out in order to determine the cyclic stress–strain behavior and the Bauschinger ratio. The in-plane anisotropy (r-value and yield stress directionalities) of these sheet materials was also compared with the results of numerical simulations using the non-associated model. These results showed that this non-associated, mixed hardening model significantly improves the prediction of earing in the cup drawing process and the prediction of springback in the sidewall of drawn channel sections, even when a simple quadratic constitutive model is used.     

FE-analysis of sheet metal forming processes using continuous contact treatment

Discrete meshes cause stepwise propagation of the contact nodes of a sheet despite the fact that the contact region in the actual forming process is altered very smoothly. This can cause problems of convergence and accuracy in contact-sensitive processes, such as a bending process. In this study, a scheme for a continuous contact treatment is proposed in order to consider the more realistic behavior of the contact phenomena during the forming process. For verification of the proposed method, the contact pressures and forming load are evaluated during the compression forming of a tube. The analysis of a hemispherical dome formed without a blank holder is also presented in order to investigate the effects of the proposed algorithm. The results show that the precise deformation mode is predicted by the utilization of the proposed method.     

Effects of texture on shear band formation in plane strain tension/compression and bending

In this study, effects of typical texture components observed in rolled aluminum alloy sheets on shear band formation in plane strain tension/compression and bending are systematically studied. The material response is described by a generalized Taylor-type polycrystal model, in which each grain is characterized in terms of an elastic–viscoplastic continuum slip constitutive relation. First, a simple model analysis in which the shear band is assumed to occur in a weaker thin slice of material is performed. From this simple model analysis, two important quantities regarding shear band formation are obtained: i.e. the critical strain at the onset of shear banding and the corresponding orientation of shear band. Second, the shear band development in plane strain tension/compression is analyzed by the finite element method. Predictability of the finite element analysis is compared to that of the simple model analysis. Third, shear band developments in plane strain pure bending of a sheet specimen with the typical textures are studied. Regions near the surfaces in a bent sheet specimen are approximately subjected to plane strain tension or compression. From this viewpoint, the bendability of a sheet specimen may be evaluated, using the knowledge regarding shear band formation in plane strain tension/compression. To confirm this and to encompass overall deformation of a bent sheet specimen, including shear bands, finite element analyses of plane strain pure bending are carried out, and the predicted shear band formation in bent specimens is compared to that in the tension/compression problem. Finally, the present results are compared to previous related studies, and the efficiency of the present method for materials design in future is 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 Al₂O₃ 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 [0₄], [90₄], 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.     

Effects of superimposed hydrostatic pressure on sheet metal formability

The effect of superimposed hydrostatic pressure on sheet metal formability is studied analytically and numerically. A tensile sample of power-law hardening material under superimposed hydrostatic pressure is first analyzed using the classical isotropic plasticity theory. It is demonstrated that the superimposed hydrostatic pressure p lowers the true tensile stress level at yielding by the amount of p, while material work-hardening is independent of p. It is showed, based on the Considère criterion, that the superimposed hydrostatic pressure increases the uniform strain. The effect of superimposed hydrostatic pressure on sheet metal formability is further assessed by constructing the Forming Limit Diagram (FLD) based on the M-K approach. It is found that the superimposed pressure delays the initiation of necking for any strain path. The difference in predicted FLDs between the superimposed hydrostatic pressure and the stress component normal to the sheet plane is demonstrated. Finally, the effect of superimposed hydrostatic pressure on fracture in round bars under tension is studied numerically using the finite element method, based on the Gurson damage model. The experimentally observed transition of the fracture surface, from the cup-cone mode under atmospheric pressure to a slant structure under high pressure, is numerically reproduced.     

Grain-boundary sliding and separation in polycrystalline metals: application to nanocrystalline fcc metals

In order to model the effects of grain boundaries in polycrystalline materials we have coupled a crystal-plasticity model for the grain interiors with a new elastic-plastic grain-boundary interface model which accounts for both reversible elastic, as well irreversible inelastic sliding-separation deformations at the grain boundaries prior to failure. We have used this new computational capability to study the deformation and fracture response of nanocrystalline nickel. The results from the simulations reflect the macroscopic experimentally observed tensile stress-strain curves, and the dominant microstructural fracture mechanisms in this material. The macroscopically observed nonlinearity in the stress-strain response is mainly due to the inelastic response of the grain boundaries. Plastic deformation in the interior of the grains prior to the formation of grain-boundary cracks was rarely observed. The stress concentrations at the tips of the distributed grain-boundary cracks, and at grain-boundary triple junctions, cause a limited amount of plastic deformation in the high-strength grain interiors. The competition of grain-boundary deformation with that in the grain interiors determines the observed macroscopic stress-strain response, and the overall ductility. In nanocrystalline nickel, the high-yield strength of the grain interiors and relatively weaker grain-boundary interfaces account for the low ductility of this material in tension.     

Dynamic plasticity and fracture in high density polycrystals: constitutive modeling and numerical simulation

Presented is a constitutive framework for modeling the dynamic response of polycrystalline microstructures, posed in a thermodynamically consistent manner and accounting for finite deformation, strain rate dependence of flow stress, thermal softening, thermal expansion, heat conduction, and thermoelastic coupling. Assumptions of linear and square-root dependencies, respectively, of the stored energy and flow stresses upon the total dislocation density enable calculation of the time-dependent fraction of plastic work converted to heat energy. Fracture at grain boundary interfaces is represented explicitly by cohesive zone models. Dynamic finite element simulations demonstrate the influences of interfacial separation, random crystallographic orientation, and grain morphology on the high-rate tensile response of a realistic two-phase material system consisting of comparatively brittle pure tungsten (W) grains embedded in a more ductile matrix of tungsten-nickel iron (W-Ni-Fe) alloy. Aspects associated with constitutive modeling of damage and failure in the homogenized material system are discussed in light of the computational results.     

Void coalescence within periodic clusters of particles

The effect of particle clustering on void damage rates in a ductile material under triaxial loading conditions is examined using three-dimensional finite element analysis. An infinite material containing a regular distribution of clustered particles is modelled using a unit cell approach. Three discrete particles are introduced into each unit cell while a secondary population of small particles within the surrounding matrix is represented using the Gurson-Tvergaard-Needleman (GTN) constitutive equations. Deformation strain states characteristic of sheet metal forming are considered; that is, deep drawing, plane strain and biaxial stretching. Uniaxial tensile stress states with varying levels of superimposed hydrostatic tension are also examined.The orientation of a particle cluster with respect to the direction of major principal loading is shown to significantly influence failure strains. Coalescence of voids within a first-order particle cluster (consisting of three particles) is a stable event while collapse of inter-cluster ligaments leads to imminent material collapse through void-sheeting.