Sheet Necking Research Articles

Heterogeneous initiation and growth of sample-scale shear bands during necking of Al-Mg sheet

Microstructural studies of Al-4.4% Mg-0.4% Mn polycrystalline sheet show that zones of intense shearing flow develop across large dimensions of microstructure during plane strain extension. These bands initiate heterogeneously at free surfaces in specific locations of a through-thickness neck. During continued sheet extension they grow in volume and move relative to a coordinate system fixed to microstructure. Geometric models are introduced to describe these spatially nonuniform processes and the sample-scale flow localization which accompanies them. There exists a geometric barrier to initiation of sample-scale shearing. Initiation takes place heterogeneously where the barrier is minimum.

Development of sample-scale shear bands during necking of ferrite-austenite sheet

This study is a quantitative, experimental study of development of shear band formation during plane strain extension of ferrite-austenite sheet. The position of aligned austenite islands was used to measure internal shape change throughout the development of flow localization to sheet failure. Through-thickness shearing flow and rotation of material by an angle φ develop during nonuniform thinning in order to maintain compatibility between deforming subelements of sheet. Surface grains fail in the locations of sheet where φ is near maximum. A pair of short shear bands form at the location of surface grain separation. Sample-scale shear bands initiate during subsequent deformation from one of the short shear bands created by surface grain failure. Sample-scale bands initiate heterogeneously and grow in volume during continuing sheet extension. Sheet failure develops once sample-scale shear bands grow across the entire cross section of sheet. These processes are described using a series of solid element models.

Geometric models of internal shape change as Shear Bands Form during Plane Strain Extension

This paper systematically presents geometric models for describing internal shape change during plane strain extension of sheet. The models describe the geometry of plastic flow localization during sheet necking and formation of sample-scale shear bands within the neck. The emphasis is toward identifying three-dimensional, solid element models that correctly describe (1) the heterogeneous initiation of simple shearing flow, inside a band that occupies just a small fraction of the sheet thickness, (2) the gradual growth of bands of simple shearing across the sheet cross section, and (3) the development of a variety of patterns of sample-scale shear bands before sheet fracture. Models are identified that agree with experimental evidence. The geometry of these models provides quantitative links between flow localization throughout a deforming sample and changes in microscopic or crystal-scale flow behavior.

Overview no. 42 Texture development and strain hardening in rate dependent polycrystals

A new rate dependent constitutive model is developed for polycrystals subjected to arbitrarily large strains. The model is used to predict deformation textures and large-strain strain hardening behavior following various stress-strain histories for single phase f.c.c. aggregates that deform by crystallographic slip. Examples involving uniaxial and plane strain tension and compression are presented which illustrate how texture influences polycrystalline strain hardening, in particular these examples demonstrate both textural strengthening and softening effects. Input to the model includes the description of single crystal strain hardening and latent hardening along with strain rate sensitivity, all properties described on the individual slip system level. The constitutive formulation used for the individual grains is essentially that developed by Peirce et al. [6, Acta metall. 31, 1951 (1983)] to solve rate dependent boundary value problems for finitely deformed single crystals. Inclusion of rate dependence is shown to overcome the long standing problem of nonuniqueness in the choice of active slip systems which is inherent in the rate independent theory. Because the slipping rates on all slip systems within each grain are unique in the rate dependent theory, the lattice rotations and thus the textures that develop are unique. In addition, the model makes it possible to study how strain rate sensitivity on the slip system, and single grain, levels is manifested in polycrystalline strain rate sensitivity. The model is also used to predict “constant offset plastic strain yield surfaces” for materials that are nearly rate insensitive—these calculations describe the development of rounded “yield surface vertices” and the resulting softening of material stiffness to a change in loading path that vertices imply. For our rate dependent solid this reduction in stiffness occurs after small but finite loading increments. Finally the model is used to carry out an imperfection-based sheet necking analysis both for isotropic and strongly textured sheets. The results show that larger strain hardening rates, and strain rate sensitivity, on the slip system level both increase the failure strains, as expected, but also demonstrate a strong influence of texture on localized necking.

1982 Annual Technical Report: Information Sciences Institute, University of Southern California, ISI /SR.83.23, Information Sciences Institute, University of Southern California, CA 90291-6695, USA (1982)

Abstract : Progress is reviewed for a research program which has examined the multiaxial deformation and fracture behavior of Ti alloys, particulary as influenced by impurities such as hydrogen and by microstructural defects such as porosity. The studies range from experimental and analytical modeling of critical fracture processes to structure-property studies of both wrought material in the form of sheet and powder fabricated material in bar form. Specifically the program has studied: (1) the modeling of ductile fracture through an examination of the deformation and fracture at isolated holes in plane-strain tension; (2) the deformation and fracture of P/M Ti and Ti-6A1-4V containing porosity; (3) the influence of hydrogen on the multiaxial deformation and fracture behavior of beta-phase (bcc) Ti-30V and alpha (hcp)-beta Ti-6A1-4V; and (4) the theory of localized necking of sheet at negative minor strains.

Localized Necking of Sheet at Negative Minor Strains

Localized necking in sheet metal has been examined for strain paths between uniaxial tension and plane strain (i.e., the negative minor strain region of a forming limit diagram). The behavior of sheet with preexisting imperfections has been analyzed and is contrasted to that free of imperfections. In particular, it is shown that the size and orientation of an imperfection is critical in determining whether or not localized necking is initiated along the imperfection. The influence of strain hardening, strain rate hardening, and plastic anisotropy on localized necking of an imperfect sheet is also examined. One of the most significant conclusions obtained from present analysis and from a reexamination of Hill’s theory is the prediction of a critical thickness strain criterion for the onset of localized necking at negative minor strains, regardless of whether or not an imperfection is present. The critical thickness strain criterion is observed in Ti alloys, Al alloys, steels, and brass.

Effect of thermal softening on Swift and Hill instability limits in forming

AbstractSwift's theory of diffuse necking and Hill's theory of localized necking in sheets subjected to in-plane biaxial stresses have been generalized to take into account thermal effects. Using a characteristic material constant M which combines the density, specific heat, and rate of thermal softening of the material, it is shown that adiabatic conditions correspond to the M-value approaching zero. At the other extreme, isothermal conditions correspond to an M-value approaching infinity. Practically, a material such as titanium, with a low M-value (M ≤ 2), will be particularly prone to thermally assisted shear instability, whereas materials such as copper, with high M-values, will be resistant to thermally assisted shear instability except at high rates of strain. As the M-value decreases, it is shown that the Swift and Hill instability loci (plotted in terms of principal strains) shrink towards the origin. Thermally assisted shear mechanisms will affect other limit situations for which the Swift or Hi...

Localized necking in thin sheets

B y using a simplified constitutive model of a pointed vertex on subsequent yield loci, namely, such that the equations of deformation-theory of rigid-plastic solids apply for fully-active stress increments, the onset of localized necking under biaxial stretching has been predicted. The predictions agree reasonably well with reported experimental observations. Since localized necking under biaxial stretching of a uniform and homogeneous sheet is impossible when flow theories of plasticity with smooth yield-loci are used, this result supports the hypothesis of vertex-formation on the yield locus under continued plastic flow. The implications of this conclusion with respect to the study of the inception of ductile fracture in solids, viewed as a material instability, may be far-reaching. Still, explanations based on a smooth yield-locus but small initial inhomogeneities cannot be ruled out, and both initial imperfections and yield-vertex effects may contribute in general to localization instabilities.

On discontinuous plastic states, with special reference to localized necking in thin sheets

Permissible discontinuities of stress, velocity, and surface slope are investigated in a plastic-rigid sheet deformed in its plane. One such discontinuity of velocity is shown to be the mathematical idealization of localized necking; the necessary restrictions on the stress-state and rate of workhardening are obtained for any yield function and plastic potential. The results are illustrated by an examination of the modes of necking in notched tension strips. The constraint factors at the yield point are obtained for notches with wedge-shaped or circular roots.