Strain gradient crystal plasticity effects on flow localization

In metal grains one of the most important failure mechanisms involves shear band localization. As the band width is small, the deformations are affected by material length scales. To study localization in single grains a rate-dependent crystal plasticity formulation for finite strains is presented for metals described by the reformulated Fleck–Hutchinson strain gradient plasticity theory. The theory is implemented numerically within a finite element framework using slip rate increments and displacement increments as state variables. The formulation reduces to the classical crystal plasticity theory in the absence of strain gradients. The model is used to study the effect of an internal material length scale on the localization of plastic flow in shear bands in a single crystal under plane strain tension. It is shown that the mesh sensitivity is removed when using the nonlocal material model considered. Furthermore, it is illustrated how different hardening functions affect the formation of shear bands.     

Flow localization in the plane strain tensile test

The deformations in a plane strain tensile test are analyzed numerically, both for a solid characterized by a phenomenological corner theory of plasticity and for a nonlinear elastic solid. As opposed to the simplest flow theory of plasticity with a smooth yield surface, both these material models exhibit shear band instabilities at a realistic level of strain. Initial imperfections are specified in the form of thickness inhomogeneities. A long-wavelength imperfection grows into the well-known necking mode and subsequently, at a sufficiently high local strain level, bands of intense shear deformations develop in the necking region. The location of these shear bands is strongly influenced by the location of small strain concentrations near the surface, induced by various short-wave patterns of initial thickness imperfections. In accord with the non-uniform straining in the neck it is found that the intensity of the localized deformations varies along the bands, and some of the shear bands end inside the material.     

Plastic spin associated with a non-normality theory of plasticity

A plasticity model using a vertex-type plastic flow rule on a smooth yield surface for an anisotropic solid has been proposed recently. This model is here completed by incorporating the effect of plastic spin. Simple shear with a large shear strain is one of the hardest tests on finite strain anisotropic plasticity models, and it is here shown which plastic spin expression is needed to avoid unrealistic oscillatory behavior of the shear stress under large shear strains. The idea of using non-normality with a smooth yield surface originates from a recent proposal of using an abrupt strain path change to determine the subsequent yield surface shape. For this method both polycrystal plasticity calculations and experiments have shown a vertex-type response on the apparently smooth yield surface.     

Interface crack growth for anisotropic plasticity with non-normality effects

A plasticity model with a non-normality plastic flow rule is used to analyze crack growth along an interface between a solid with plastic anisotropy and an elastic substrate. The fracture process is represented in terms of a traction-separation law specified on the crack plane. A phenomenological elastic–viscoplastic material model is applied, using an anisotropic yield criterion, and in each case analyzed the effect of non-normality is compared with results for the standard normality flow rule. Due to the mismatch of elastic properties across the interface the corresponding elastic solution has an oscillating stress singularity, and with conditions of small scale yielding this solution is applied as boundary conditions on the outer edge of the region analyzed. Crack growth resistance curves are calculated numerically, and the effect of the near-tip mode mixity on the steady-state fracture toughness is determined. It is found that the steady-state fracture toughness is quite sensitive to differences in the initial orientation of the principal axes of the anisotropy relative to the interface.     

Crack growth resistance for anisotropic plasticity with non-normality effects

For a plastically anisotropic solid a plasticity model using a plastic flow rule with non-normality is applied to predict crack growth. The fracture process is modelled in terms of a traction–separation law specified on the crack plane. A phenomenological elastic–viscoplastic material model is applied, using one of two different anisotropic yield criteria to account for the plastic anisotropy, and in each case the effect of the normality flow rule is compared with the effect of non-normality. Conditions of small scale yielding are assumed, with mode I loading conditions far from the crack-tip, and various directions of the crack plane relative to the principal axes of the anisotropy are considered. It is found that the steady-state fracture toughness is significantly reduced when the non-normality flow rule is used. Furthermore, it is shown that the predictions are quite sensitive to the value of the maximum angle of deviation from normality in the non-normality flow rule.     

Effect of yield surface curvature and void nucleation on plastic flow localization

The effect of void nucleation is incorporated in a recently proposed material model that accounts for a combination of kinematic hardening and isotropic hardening of a porous ductile material. Since each of plastic dilatancy, void nucleation and yield surface curvature have a strong influence on predictions of plastic flow localization, the present material model can be used to study the interaction of these effects. Nucleation controlled by the plastic strain as well as nucleation controlled by the maximum normal stress on the particle-matrix interface are modelled. The predictions of the material model, for various combinations of parameters, are illustrated by analyses of shear band formation under plane strain or axisymmetric conditions, and by analyses of necking in biaxially stretched sheets.     

Piobert–Lüders plateau and Portevin–Le Chatelier effect in an Al–Mg alloy in simple shear

The jerky flow in an Al–Mg alloy is studied during simple shear tests at room temperature and various strain rates. Direct observations of the sample surface using digital image correlation allow the study of the type and the dynamics of bands associated to plastic instabilities as a function of shear strain and shear strain rate. The paper features that both Piobert–Lüders and Portevin–Le Chatelier phenomena can be observed for a simple shear stress state at room temperature. The nucleation, growth and movement of the bands are described: it is shown that the kinematics of the bands is similar to those observed in tension but that the orientation of the bands varies with the shear strain.     

Hydrogen induced shear localization of the plastic flow in metals and alloys

Hydrogen enhanced localized plasticity (HELP) is a viable mechanism for hydrogen embrittlement supported by experimental observations. According to the HELP mechanism, hydrogen induced premature failures result from hydrogen induced plastic instability which leads to hydrogen assisted localized ductile processes. The objective of this work is to reveal the role of hydrogen in possibly localizing the macroscopic deformation into bands of intense shear using solid mechanics methodology. The hydrogen effect on material deformation is modeled through the hydrogen induced volume dilatation and the reduction in the local flow stress upon hydrogen dissolution into the lattice. Hydrogen in assumed to reside in both normal interstitial lattice sites (NILS) and reversible traps associated with the plastic deformation. The analysis of the plastic deformation and the conditions for plastic flow localization are carried out in plane strain uniaxial tension. For a given initial hydrogen concentration in the unstressed specimen, a critical macroscopic strain is identified at which shear localization commences.     

Granular plasticity,a contribution from discrete mechanics

Failures by divergence instabilities in rate-independent non-associated material, such as granular matter, can occur from mechanical states described by the plastic stress limit surface, but also from stress states strictly included within this surface. Besides, the failure mode may be localized, with for instance the formation of shear bands, or diffuse with a strain field remaining homogeneous. All these failures can be described in a unique framework where plastic limit stress states appear as particular cases of generalized limit states; and where the effective development of failure is characterized by the unbounded increase of response parameters linked by a failure rule (i.e. a generalized plastic flow rule), together with a bifurcation of the mechanical response from a quasi-static pre-failure response to a dynamic post-failure one. All these features are discussed and highlighted from direct numerical simulations performed with a discrete element model. Moreover, the second order work criterion directly related at the macroscopic scale to divergence instabilities, is shown to be also relevant at the scale of inter-particle contacts.     

Post-critical plastic deformation in incrementally nonlinear materials

The formation of multiple macroscopic shear bands is investigated as a mechanism of advanced plastic flow of polycrystalline metals. The overall deformation pattern and material characteristics are determined beyond the critical instant of ellipticity loss, without the need of introducing an internal length scale. This novel approach to the modelling of post-critical plastic deformation is based on the concept of a representative nonuniform solution in a homogeneous material. The indeterminacy of a post-critical representative solution is removed by eliminating unstable solution paths with the help of the energy criterion of path instability. It is shown that the use of micromechanically based, incrementally nonlinear corner theories of time-independent plasticity leads then to gradual concentration of post-critical plastic deformation. The volume fraction occupied by shear bands is found to have initially a well-defined, finite value insensitive to the mesh size in finite element calculations. Further deformation depends qualitatively on details of the constitutive law. In certain cases, the volume fraction of active bands decreases rapidly to zero, leading to material instability of dynamic type. However, for physically hardening materials with the yield-vertex effect, the localization volume typically remains finite over a considerable deformation range. At later stages of the plane strain simulation, differently aligned secondary bands are formed in a series of bifurcations.     

An analysis of ductile rupture in notched bars

Ductile fracture in axisymmetric and plane strain notched tensile specimens is analyzed numerically, based on a set of elastic-plastic constitutive relations that account for the nucleation and growth of microvoids. Final material failure by void coalescence is incorporated into the constitutive model via the dependence of the yield function on the void volume fraction. In the analyses the material has no voids initially; but as the voids nucleate and grow, the resultant dilatancy and pressure sensitivity of the macroscopic plastic flow influence the solution significantly. Considering both a blunt notch geometry and a sharp notch geometry in the computations permits a study of the relative roles of high strain and high triaxiality on failure. Comparison is made with published experimental results for notched tensile specimens of high-strength steels. All axisymmetric specimens analyzed fail at the center of the notched section, whereas failure initiation at the surface is found in plane strain specimens with sharp notches, in agreement with the experiments. The results for different specimens are used to investigate the circumstances under which fracture initiation can be represented by a single failure locus in a plot of stress triaxiality vs effective plastic strain.     

Non-normality and bifurcation in plane strain tension and compression

The bifurcations of a rectangular block subject to plane strain tension or compression are investigated. The block material is taken to be incompressible and is characterized by an incrementally linear constitutive law for which “normality” does not necessarily hold. The consequences of non-normality regarding bifurcation are given primary emphasis here. The characteristic regimes of the governing equations (elliptic, parabolic and hyperbolic) are detennined. In each of these regimes both symmetric and antisymmetric diffuse bifurcation modes are available. Additionally, in the hyperbolic and parabolic regimes, bifurcation into a localized shear band mode is also possible. Particular attention is given to the limiting cases of long wavelength and soon wavelength diffuse bifurcation modes. The range of parameter values is identified for which bifurcation into some localized mode may precede bifurcation into a long wavelength diffuse mode. Some difficulties associated with employing a linear incremental solid in a bifurcation analysis, when primary interest is in the bifurcation of an underlying elastic-plastic solid, are also discussed.     

金属材料平面应变拉伸变形局部化有限元分析

本文对于涉及韧性金属大变形中颈缩与剪切带断裂一类高度非线性变形局部化问题进行了弹塑性有限元数值模拟。采用改进的Ｊ２形变理论微分形式公式与交叉三角形四边形单元有限元网格，详细研究了应变硬化指数及初始表面不均匀特性的平面应变拉伸颈缩和剪切带形成的综合影响，给出此类问题的断裂机制图。     

Analysis of anisotropy and plastic spin effects on localization phenomena

Summary The main objective of the paper is the investigation of the influence of the anisotrophy and plastic spin effects on criteria for adiabatic shear band localization of plastic deformation. A theory of thermoplasticity is formulated within a framework of the rate-type covariance material structure with a finite set of internal state variables. The theory takes into consideration such effects as plastic non-normality, plastic-induced anisotropy (kinematic hardening), micro-damage mechanism, thermomechanical coupling and plastic spin. The next objective of the paper is to focus attention on cooperative phenomena in presence of the plastic spin, and the discussion on the influence of synergetic effects on localization criteria. A particular constitutive law for the plastic spin is assumed. The necessary condition for a localized plastic deformation region to be formed is obtained. This condition is accomplished by the assumption that some eigenvalues of the instantaneous adiabatic acoustic tensor vanish. A procedure has been developed which allows us to discuss two separate groups of effects on the localization phenomenon along a shear band. Plastic spin, spatial covariance and kinematic hardening effects are investigated at an isothermal process in an undamaged solid. In the second case, an adiabatic process in a damaged solid is discussed when the spatial covariance terms and the plastic spin are neglected. Here the thermomechanical coupling, micro-damage mechanism and kinematic hardening effects are examined. For both cases, the criteria for adiabatic shear band localization are obtained in an exact analytical form. Particular attention is focused on the analysis of the following effects: (i) plastic non-normality; (ii) plastic spin; (iii) covariant terms; (iv) plastic strain-induced anisotropy; (v) micro-damage mechanism; (vi) thermomechanical couplings. Cooperative phenomena are considered, and synergetic effects are investigated. A discussion of the influence of the plastic spin, kinematic hardening and covariant terms on the shear band localization conditions is presented. A numerical estimation of the effects discussed is given. Received 24 April 1997; accepted for publication 23 December 1997     

Adiabatic shear banding and scaling laws in chip formation with application to cutting of Ti–6Al–4V

The phenomenon of adiabatic shear banding is analyzed theoretically in the context of metal cutting. The mechanisms of material weakening that are accounted for are (i) thermal softening and (ii) material failure related to a critical value of the accumulated plastic strain. Orthogonal cutting is viewed as a unique configuration where adiabatic shear bands can be experimentally produced under well controlled loading conditions by individually tuning the cutting speed, the feed (uncut chip thickness) and the tool geometry. The role of cutting conditions on adiabatic shear banding and chip serration is investigated by combining finite element calculations and analytical modeling. This leads to the characterization and classification of different regimes of shear banding and the determination of scaling laws which involve dimensionless parameters representative of thermal and inertia effects. The analysis gives new insights into the physical aspects of plastic flow instability in chip formation. The originality with respect to classical works on adiabatic shear banding stems from the various facets of cutting conditions that influence shear banding and from the specific role exercised by convective flow on the evolution of shear bands. Shear bands are generated at the tool tip and propagate towards the chip free surface. They grow within the chip formation region while being convected away by chip flow. It is shown that important changes in the mechanism of shear banding take place when the characteristic time of shear band propagation becomes equal to a characteristic convection time. Application to Ti–6Al–4V titanium are considered and theoretical predictions are compared to available experimental data in a wide range of cutting speeds and feeds. The fundamental knowledge developed in this work is thought to be useful not only for the understanding of metal cutting processes but also, by analogy, to similar problems where convective flow is also interfering with adiabatic shear banding as in impact mechanics and perforation processes. In that perspective, cutting speeds higher than those usually encountered in machining operations have been also explored.     

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.     

Quasi-static crush behavior of aluminum honeycomb specimens under compression dominant combined loads

The influence of shear stress on the quasi-static crush behavior of aluminum honeycomb specimens under compression dominant combined loads is investigated by experiments. A test fixture was designed such that dominant compressive and shear loads with respect to the strongest material symmetry direction can be controlled and applied independently. Honeycomb specimens were also designed such that the secondary non-uniform stresses due to the stress-free boundary can be minimized. The experimental results indicate that the normal crush strengths under combined compressive and shear loads are lower than that under pure compressive loads. A phenomenological yield criterion for specimens with different in-plane orientation angles is proposed based on the experimental normal crush and shear strengths under combined loads. The experimental results suggest non-normality plastic flow based on the yield criterion. The non-normality flow behavior becomes more pronounced as the in-plane orientation angle increases. The experimental results also indicate that the energy absorption rate depends upon the ratio of the shear stress to the compressive stress and the in-plane orientation angle. In addition, specimens crushed under combined loads show inclined stacking patterns of folds due to the asymmetric location of horizontal plastic hinge lines and the rupture of aluminum cell walls along the adhesive lines. These experimental observations are useful to develop microscopic plasticity models of aluminum honeycombs under compression dominant combined loads.     

Instability and localization of plastic flow in shear at high strain rates

Shear band formation in a thermal viscoplastic heat conducting material is described in a simple shear test at high strain rate with inertia effects. The classical perturbation method is discussed, and a new relative perturbation method accounting for non-steadiness of plastic flow is presented. They respectively provide instability and localization criteria which are compared. Furthermore both are compared to available nonlinear exact results and to experimental data. The influence of material parameters, initial imperfections, and boundary conditions is described.     

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.     

A scaling law for the effect of inertia on the formation of adiabatic shear bands

For a rigid/perfectly plastic material with linear thermal softening and power law rate hardening there is a competition between heat conduction and inertia in determining the time of shear band formation. In a finite specimen the nominal strain rate that produces the fastest growth of perturbations corresponds to the minimum critical strain. Similarly for a fixed strain rate in an infinite specimen, there is a finite wavelength with the maximum growth rate. It is argued that this wavelength should correspond to the most probable minimum spacing for shear bands.