The emergence of shear bands in plane strain

In this paper, we examine the effect of a slight material imperfection on the deformation field in an otherwise homogeneous body subjected to a plane bi-axial stretch at infinity. Both hyper-elastic and hypo-elastic materials are considered, with the constitutive equations assumed to be such that the governing equilibrium equations lose ellipticity at some strain level. A straightforward regular perturbation analysis is performed and attention is focussed on the features of the first order terms. It is found that the effect of the imperfection is negligible at small values of the applied load. As the load increases, the imperfection more or less abruptly gets “activated” and causes a rapid concentration of strain within certain narrow bands—shear bands—passing through the imperfection. In order to estimate the accuracy of the linearized analysis, a finite element solution of the nonlinear problem is also carried out and results are compared.     

Formation of shear bands in plane sheet

The formation of shear bands in plane sheet is studied, both analytically and experimentally, to enhance the fundamental understanding of this phenomenon and to develop a capability for predicting material failure. The evolution of voids is measured and its interaction with the process of shear banding is examined. The evolving dilatancy in plasticity is shown to have a vital role in analysing the shear-band type of bifurcation, and tremendously reduces the theoretical value of critical stresses. The analyses, referring to both localized and diffuse modes of bifurcation, fairly explain the corresponding observations obtained through testing a dual-phase steel sheet and provide a justification of the constitutive model used.     

Initiation of localized plane deformations at a circular cavity in an infinite compressible nonlinearly elastic medium

This investigation is concerned with the plane strain deformation of an infinite slab, containing a circular cavity, within the theory of finite elastostatics for a particular homogeneous isotropic compressible material, the so-called Blatz-Ko material. The body is subjected to uniform pressure, either internal or external. Exact closed-form solutions for the axisymmetric deformation and stress fields are obtained. In the case of internal pressure, it is found that the applied pressure may not exceed a certain maximum value p _max. At a value of pressure p _e (<p _max), the governing equations lose ellipticity at the cavity wall. For greater values of pressure this solution remains smooth, though involving both elliptic and non-elliptic regions. Non-existence of axisymmetric solutions with discontinuous strain fields is established. The possibility of bifurication into a surface mode is considered and it is shown that this occurs at a value of pressure slightly smaller than p _e. Such surface wrinkling leads to a periodic distribution of points of stress concentration, from which shear bands may initiate.This work was supported by the U.S. Army Research Office under Grant DAAG29-83-K-0145 (R.A. & C.O.H.) and by the U.S. National Science Foundation under Grant MEA 78-26071 (C.O.H.).     

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.     

On stress collapse in adiabatic shear bands

The dynamics of adiabatic shear band formation is considered making use of a simplified thermo/visco/plastic flow law. A new numerical solution is used to follow the growth of a perturbation from initiation, through early growth and severe localization, to a slowly varying terminal configuration. Asymptotic analyses predict the early and late stage patterns, but the timing and structure of the abrupt transition to severe localization can only be studied numerically, to date. A characteristic feature of the process is that temperature and plastic strain rate begin to localize immediately, but only slowly, whereas the stress first evolves almost as if there were no perturbation, but then collapses rapidly when severe localization occurs.     

Structure of adiabatic shear bands in thermo-viscoplastic materials

The shear band structure in a thermo-viscoplastic material has been analyzed under quasi-static conditions during the final stage of the localization process. A closed form expression of the bandwidth, of the size of the heat-affected zone and of the failure time has been obtained for a homogeneous material. The analysis is based on the assumption that the strain-rate distribution is steady beyond a certain strain level. Predictions have been compared with experimental measurements.     

The growth of shear bands in composite microstructures

Shear band formation in materials with inhomogeneous and composite microstructures is influenced by factors that usually do not come into play in monolithic materials. Experiments and calculations have shown that inhomogeneities in material properties enhance the localization of deformation. This investigation concerns the propagation of shear bands in a two-phase tungsten composite under the conditions of nominally pure shear deformation. Finite element calculations are carried out to delineate the effects of different grain–matrix morphologies. In the numerical models, the initiation of shear bands is triggered by a notch, simulating the effect of defects such as microcracks and microvoids in materials. Calculations demonstrate that phase morphology, particle size and the relative location of initiation site have significant influences on the development of localized deformation. The work and energy evolutions are tracked for each constituent phase in the microstructures. In addition, the exchange of thermal energy through heat flow between the phases is analyzed. The results show that a strong correlation exists between the course of shear band propagation and the thermomechanical coupling between microscopic phases.     

Dislocation structure of shear bands in single crystals

A mathematical model is proposed for the development of a shear band in crystals. The model is based on the mechanism of double cross-slips of screw-dislocation segments. Equations are derived to study instability of the uniform distribution of dislocations. A solution is found in the form of a traveling wave, which describes the shear-band structure. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 6, pp. 102–113, November–December, 2006.     

The formation of multiple adiabatic shear bands

In a previous paper, Zhou et al. [2006. A numerical methodology for investigating adiabatic shear band formation. J. Mech. Phys. Solids, 54, 904-926] developed a numerical method for analyzing one-dimensional deformation of thermoviscoplastic materials. The method uses a second order algorithm for integration along characteristic lines, and computes the plastic flow after complete localization with high resolution and efficiency. We apply this numerical scheme to analyze localization in a thermoviscoplastic material where multiple shear bands are allowed to form at random locations in a large specimen. As a shear band develops, it unloads neighboring regions and interacts with other bands. Beginning with a random distribution of imperfections, which might be imagined as arising qualitatively from the microstructure, we obtain the average spacing of shear bands through calculations and compare our results with previously existing theoretical estimates. It is found that the spacing between nucleating shear bands follows the perturbation theory due to Wright and Ockendon [1996. A scaling law for the effect of inertia on the formation of adiabatic shear bands. Int. J. Plasticity 12, 927-934], whereas the spacing between mature shear bands is closer to that predicted by the momentum diffusion theory of Grady and Kipp [1987. The growth of unstable thermoplastic shear with application to steady-wave shock compression in solids. J. Mech. Phys. Solids 35, 95-119]. Scaling laws for the dependence of band spacing on material parameters differ in many respects from either theory.     

Size-effects in plane strain sheet-necking

A finite strain generalization of the strain gradient plasticity theory by Fleck and Hutchinson (J. Mech. Phys. Solids 49 (2001a) 2245) is proposed and used to study size effects in plane strain necking of thin sheets using the finite element method. Both sheets with rigid grips at the ends and specimens with shear free ends are analyzed. The strain gradient plasticity theory predicts delayed onset of localization when compared to conventional theory, and it depresses deformation localization in the neck. The sensitivity to imperfections is analyzed as well as differently hardening materials.     

Adiabatic shear banding in plane strain tensile deformations of 11 thermoelastoviscoplastic materials with finite thermal wave speed

We study the initiation and propagation of adiabatic shear bands (ASBs) in 11 homogeneous materials each modeled as microporous, isotropic and thermoelastoviscoplastic, and deformed in plane strain tension. The heat conduction in each material is assumed to be governed by a hyperbolic heat equation; thus thermal and mechanical waves propagate with finite speeds. The decrease in the thermophysical parameters due to the increase in porosity is considered. An ASB is assumed to initiate at a material point when the maximum shear stress there has dropped to 80% of its peak value for that material point and it is deforming plastically. An approximate solution of the coupled nonlinear partial differential equations subject to suitable initial and boundary conditions is found by the finite element method (FEM). In contrast to the Considerè and the Hart criterion, it is found that an ASB initiates when the axial load drops rapidly and not when it peaks. The refinement of the 40 × 40 uniform FE mesh to 120 × 120 uniform elements decreased the ASB initiation time by 2.1% while increasing the CPU time by a factor of ∼26. By locating points where the ASB has initiated we find its current length, width and speed. The 11 materials are ranked according to the time of initiation of an ASB under otherwise identical geometric and loading conditions with the same initial nonuniform porosity distribution. This ranking of materials is found to differ somewhat from that ascertained by Batra and Kim (1992) who studied simple shearing deformations, and by Batra et al. (1995) who analyzed three-dimensional torsional deformations of thin-walled tubular specimens. The average axial strain determined from the maximum axial load condition differs noticeably from that when an ASB initiates.     

Influence of damage on properties of strain localization in geomaterials at plane stress and plane strain

Summary By regarding geomaterials under loading as a mixture of intact and damaged parts, we investigate the influence of damage on the properties of strain localization in elastoplastic geomaterials at plane stress and plane strain. Conditions for the onset of strain localization including the effects of damage are derived for the cases of plane strain and plane stress. Discussed are the inclination of the localized band and the hardening modulus corresponding to the onset of strain localization. It is shown that the properties of the strain localization are dependent on the damage and the capacity of bearing hydrostatic pressure by the damaged part, and that damage may induce an earlier onset of strain localization and lead to instability of a geomaterial.accepted for publication 11 March 2004     

The initiation and growth of adiabatic shear bands

A simple version of thermo/viscoplasticity theory is used to model the formation of adiabatic shear bands in high rate deformation of solids. The one dimensional shearing deformation of a finite slab is considered. For the constitutive assumptions made in this paper, homogeneous shearing produces a stress/strain response curve that always has a maximum when strain and rate hardening, plastic heating, and thermal softening are taken into account. Shear bands form if a perturbation is added to the homogeneous fields just before peak stress is obtained with these new fields being used as initial conditions. The resulting initial/boundary value problem is solved by the finite element method for one set of material parameters. The shear band grows slowly at first, then accelerates sharply, until finally the plastic strain rate in the center reaches a maximum, followed by a slow decline. Stress drops rapidly throughout the slab, and the central temperature increases rapidly as the peak in strain rate develops.