Microstructural effects on shear instability and shear band spacing

A constitutive relation that accounts for the thermally activated dislocation motion and microstructure interaction is used to study the stability of a homogeneous solution of equations governing the simple shearing deformations of a thermoviscoplastic body. An instability criterion and an upper bound for the growth rate of the infinitesimal deformations superimposed on the homogeneous solution are derived. By adopting Wright and Ockendon's postulate, i.e., the wavelength of the dominant instability mode with the maximum growth rate determines the minimum spacing between shear bands, the shear band spacing is computed. The effect of the initial dislocation density, the nominal strain-rate, and parameters describing the initial thermal activation and the initial microstructure interaction on the shear band spacing are delineated.     

Effect of viscoplastic relations on the instability strain,shear band initiation strain,the strain corresponding to the minimum shear band spacing,and the band width in a thermoviscoplastic material

We study thermomechanical deformations of a steel block deformed in simple shear and model the thermoviscoplastic response of the material by four different relations. We use the perturbation method to analyze the stability of a homogeneous solution of the governing equations. The smallest value of the average strain for which the perturbed homogeneous solution becomes unstable is called the critical strain or the instability strain. For each one of the four viscoplastic relations, we investigate the dependence upon the nominal strain-rate of the critical strain, the shear band initiation strain, the shear band spacing and the band width. It is found that the qualitative responses predicted by the Wright–Batra, Johnson–Cook and the power law relations are similar but these differ from that predicted by the Bodner–Partom relation. The computed band width is found to depend upon the specimen height.     

Prediction of the initial thickness of shear band localization based on a reduced strain gradient theory

Plastic flow localization in ductile materials subjected to pure shear loading and uniaxial tension is investigated respectively in this paper using a reduced strain gradient theory, which consists of the couple-stress (CS) strain gradient theory proposed by Fleck and Hutchinson (1993) and the strain gradient hardening (softening) law (C–W) proposed by Chen and Wang (2000). Unlike the classical plasticity framework, the initial thickness of the shear band and the strain rate distribution in both cases are predicted analytically using a bifurcation analysis. It shows that the strain rate is obviously non-uniform inside the shear band and reaches a maximum at the center of the shear band. The initial thickness of the shear band depends on not only the material intrinsic length l_cs but also the material constants, such as the yield strength, ultimate tension strength, the linear hardening and softening shear moduli. Specially, in the uniaxial tension case, the most possible tilt angle of shear band localization is consistent qualitatively with the existing experimental observations. The results in this paper should be useful for engineers to predict the details of material failures due to plastic flow localization.     

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.     

On shear band formation: I. Constitutive relationship for a dual yield model

A phenomenological constitutive relation, for capturing the shear band formation in a rate-independent elastic-plastic material, is established. The model takes into account both the J₂-isotropic flow and a threshold shear stress-based flow. The elastic-plastic constitutive tensor is expressed explicity in terms of elastic constants, the deviatoric stress tensor, the direction of the principal shear velocity-strain, and other material constants. This model particularly facilitates the resolution of the formation of the shear band even under material hardening conditions and does not demand an a priori knowledge of the orientation of the shear band. This is incorporated in an FEM, and the plane strain tensile test of Anand and Spitzig [1980] is numerically simulated. The computed results compare favorably with the experimental data. The shear band emerges more naturally as a solution to the boundary value problem, unlike the situations in solutions based on classical bifurcation methods. Nevertheless, the usefulness of the local instability condition (Ortiz et al. [1987]) is also demonstrated.     

Analysis of failure modes in an impact loaded thermoviscoplastic prenotched plate

Kalthoff observed experimentally that the failure mode in a prenotched maraging steel plate impact loaded on the notched side changes from brittle to ductile with an increase in the impact speed. Here we numerically investigate the effect on the failure mode transition speed of the shape of the notch-tip and the presence of a hole ahead of a circular notch-tip. The shape of the notch-tip is varied by changing the ratio, a/b, of the principal axes of an elliptic notch. For a circular notch-tip, we also investigate the effect, on the failure mode transition speed, of the presence of a circular void ahead of the notch-tip and situated either on or away from the axis of the notch. The Bodner–Partom thermoviscoplastic relation is used to model the strain hardening, strain-rate hardening and thermal softening response of the material of the plate. The transient plane strain thermomechanical deformations of the plate are analyzed by the finite element method. It is found that for a/b=2.0 and 10.0, the brittle failure preceded the ductile failure for the six impact speeds studied herein, and for a/b=0.4 and 1.0, the failure mode transitioned from the brittle to the ductile with an increase in the impact speed. The presence of a circular void ahead of the circular notch-tip shifts towards the axis of the notch the point on the the notch-tip surface where a shear band initiates.     

Analysis of dynamic shear bands in an FCC single crystal

We study plane strain dynamic thermomechanical deformations of an fcc single crystal compressed along the crystallographic direction [010] at an average strain rate of 1000 sec⁻¹. Two cases are studied; one in which the plane of deformation is parallel tothe plane (001) of the single crystal, and another one with deformation occuring in the plane (101̄) of the single crystal. In each case, the 12 slip systems are aligned symmetrically about the two centroidal axes. We assume that the elastic and plastic deformations of the crystal are symmetrical about these two axes. The crystal material is presumed to exhibit strain hardening, strain-rate hardening, and thermal softening. A simple combined isotropic-kinematic hardening expression for the critical resolved shear stress, proposed by Weng, is modified to account for the affine thermal softening of the material. When the deformation is in the plane (001) of the single crystal, four slip systems (111)[11̄0], (111̄)[11̄0], (11̄;1̄;)[110], and (11̄1)[110] are active in the sense that significant plastic deformations occur along these slip systems. However, when the plane of deformation is parallel to the plane (101̄;) of the single crystal, slip systems (11̄;1)[110], (11̄1)[011], (111)[11̄0], and (111)[01̄1] are more active than the other eight slip systems. At an average strain of 0.0108, the maximum angle of rotation of a slip system within a shear band, about an axis perpendicular to the plane of deformation, is found to be 20.3° in the former case, and 22.9° in the latter.     

Dynamic limit analysis formulation for impact simulation of structural members

This paper introduces an extended concept of limit analysis to deal with the dynamic equilibrium condition considering the inertia and strain-rate effect for dynamic behavior of structures. The conventional limit analysis method has been applied to only static collapse analysis of structures without consideration of dynamic effects in the structural behavior. A dynamic formulation for the limit analysis has been derived for incremental analysis dealing with time integration, strain and stress evaluation, strain hardening, strain-rate hardening and thermal softening. The time dependent term in the governing equation is integrated with the WBZ-α method. The dynamic material behavior is described by the Johnson–Cook model in order to consider strain-rate hardening and thermal softening as well as strain hardening. Simulations have been carried out for impact analysis of a Taylor bar and an S-rail and their numerical results are compared with elasto-plastic explicit analysis results by LS-DYNA3D. Comparison demonstrates that the dynamic finite element limit analysis can predict the crashworthiness of structural members effectively with less effort and computing time than the commercial code compared. The crashworthiness of a structure with the rate-dependent constitutive model is also compared to that with the quasi-static constitutive relation in order to investigate the dynamic effect on deformation of structures.     

Shear band formation in thermo-viscoplastic solids under plane strain compression

The plane strain compression of a rectangular block is numerically investigated for the study of dynamic shear band development in thermo-elasto-viscoplastic materials from an internal inhomogeneity. As expected, it plays an important role in triggering the onset of shear, localization as well as thermal softening. And the competition between the strain, strain-rate hardening and thermal softening exists throughout the process. It is found that shear band develops at a 45-degree angle to the compression axis. In the light of given patterns of deformation and temperature, shear band evolution accelerated by thermal softening is retarded by the inertial effects. Interestingly, a similar temperature band is also formed along the trajectory of the localized deformation band. The calculations also show the energy evolution during the coupled thermo-mechanical process of shear band propagation. Finally, the mesh effect is discussed in terms of the numerical results from two different meshes. The project is supported by the National Natural Sciences Foundation of China.     

Asymptotic crack-tip fields for dynamic crack growth in compressive power-law hardening materials

The effect of material compressibility on the stress and strain fields for a mode-I crack propagating steadily in a power-law hardening material is investigated under plane strain conditions. The plastic deformation of materials is characterized by the J₂ flow theory within the framework of isotropic hardening and infinitesimal displacement gradient. The asymptotic solutions developed by the present authors [Zhu, X.K., Hwang K.C., 2002. Dynamic crack-tip field for tensile cracks propagating in power-law hardening materials. International Journal of Fracture 115, 323–342] for incompressible hardening materials are extended in this work to the compressible hardening materials. The results show that all stresses, strains, and particle velocities in the asymptotic fields are fully continuous and bounded without elastic unloading near the dynamic crack tip. The stress field contains two free parameters σ_eq0 and s₃₃₀ that cannot be determined in the asymptotic analysis, and can be determined from the full-field solutions. For the given values of σ_eq0 and s₃₃₀, all field quantities around the crack tip are determined through numerical integration, and then the effects of the hardening exponent n, the Poisson ratio ν, and the crack growth speed M on the asymptotic fields are studied. The approximate behaviors of the proposed solutions are discussed in the limit of ν → 0.5 or n → ∞.     

Size effects in generalised continuum crystal plasticity for two-phase laminates

The solutions of a boundary value problem are explored for various classes of generalised crystal plasticity models including Cosserat, strain gradient and micromorphic crystal plasticity. The considered microstructure consists of a two-phase laminate containing a purely elastic and an elasto-plastic phase undergoing single or double slip. The local distributions of plastic slip, lattice rotation and stresses are derived when the microstructure is subjected to simple shear. The arising size effects are characterised by the overall extra back stress component resulting from the action of higher order stresses, a characteristic length l_c describing the size-dependent domain of material response, and by the corresponding scaling law lⁿ as a function of microstructural length scale, l. Explicit relations for these quantities are derived and compared for the different models. The conditions at the interface between the elastic and elasto-plastic phases are shown to play a major role in the solution. A range of material parameters is shown to exist for which the Cosserat and micromorphic approaches exhibit the same behaviour. The models display in general significantly different asymptotic regimes for small microstructural length scales. Scaling power laws with the exponent continuously ranging from 0 to −2 are obtained depending on the values of the material parameters. The unusual exponent value −2 is obtained for the strain gradient plasticity model, denoted “curl H^p” in this work. These results provide guidelines for the identification of higher order material parameters of crystal plasticity models from experimental data, such as precipitate size effects in precipitate strengthened alloys.     

Percolation in layered media — A conductivity approach

Long square-lattice and cubic-lattice samples consisting of many layers are simulated. Within each layer, the concentration of permeable bonds is constant whereas each layer has a different concentration chosen randomly from the interval between the percolation threshold and unit concentration. The conductivity of the random resistor network corresponding to this percolation model is calculated, both parallel and perpendicular to the layers, in both two and three dimensions. For the conductivity parallel to the layers, an effective medium calculation comes within 10% of the true conductivity. For the conductivity perpendicular to the layers, percolation theory is necessary.List of Symbols G Total Conductivity in units of the conductivity of one bond - L Length of sample in units of the length of one bond - n Width of sample in units of the length of one bond - N Number of layers - p Probability that a bond conduct - p _c Percolation threshold - R Resistivity in units of the resistivity of one bond - t Percolation conductivity exponent - v Percolation correlation length exponent - Correlation length in units of the length of one bond     

Stochastic finite element nonlinear free vibration analysis of piezoelectric functionally graded materials beam subjected to thermo-piezoelectric loadings with material uncertainties

In this paper, second order statistics of large amplitude free flexural vibration of shear deformable functionally graded materials (FGMs) beams with surface-bonded piezoelectric layers subjected to thermopiezoelectric loadings with random material properties are studied. The material properties such as Young’s modulus, shear modulus, Poisson’s ratio and thermal expansion coefficients of FGMs and piezoelectric materials with volume fraction exponent are modeled as independent random variables. The temperature field considered is assumed to be uniform and non-uniform distribution over the plate thickness and electric field is assumed to be the transverse components E _z only. The mechanical properties are assumed to be temperature dependent (TD) and temperature independent (TID). The basic formulation is based on higher order shear deformation theory (HSDT) with von-Karman nonlinear strain kinematics. A C ⁰ nonlinear finite element method (FEM) based on direct iterative approach combined with mean centered first order perturbation technique (FOPT) is developed for the solution of random eigenvalue problem. Comparison studies have been carried out with those results available in the literature and Monte Carlo simulation (MCS) through normal Gaussian probability density function.     

Effects of the strain-hardening exponent on two-parameter characterizations of surface-cracks under large-scale yielding

The influence of strain hardening exponent on two-parameter J-Q near tip opening stress field characterization with modified boundary layer formulation and the corresponding validity limits are explored in detail. Finite element simulations of surface cracked plates under uniaxial tension are implemented for loads exceeding net-section yield. The results from this study provide numerical methodology for limit analysis and demonstrate the strong material dependencies of fracture parameterization under large scale yielding. Sufficient strain hardening is shown to be necessary to maintain J-Q predicted fields when plastic flow progresses through the remaining ligament. Lower strain hardening amplifies constraint loss due to stress redistribution in the plastic zone and increases the ratio of tip deformation to J. The onset of plastic collapse is marked by shape change and/or rapid relaxation of tip fields compared to those predicted by MBL solutions and thus defining the limits of J-Q dominance. A radially independent Q-parameter cannot be evaluated for the low strain hardening material at larger deformations within a range where both cleavage and ductile fracture mechanisms are present. The geometric deformation limit of near tip stress field characterization is shown to be directly proportional to the level of stress the material is capable of carrying within the plastic zone. Accounting for the strain hardening of a material provides a more adjusted and less conservative limit methodology compared to those generalized by the yield strength alone. Results from this study are of relevance to establishing testing standards for surface cracked tensile geometries.     

Strain-rate history and temperature effects on the torsional-shear behavior of a mild steel

Previous investigations on the effects of strain-rate and temperature histories on the mechanical behavior of steel are briefly reviewed. A study is presented on the influence of strain rate and strain-rate history on the shear behavior of a mild steel, over a wide range of temperature Experiments were performed on thin-walled tubular specimens of short gage length, using a torsional split-Hopkinson-bar apparatus adapted to permit quasi-static as well as dynamic straining at different temperatures. The constant-rate behavior was first measured at nominal strain rates of 10^?3 and 10³ s^?1 for ?150, ?100, ?50, 20, 200 and 400°C. Tests were then carried out, at the same temperatures, in which the strain rate was suddenly increased during deformation from the lower to the higher rate at various large values of plastic strain. The increase in rate occurred in a time of the order of 20 μs so that relatively little change of strain took place during the jump. The low strain-rate results show a well-defined elastic limit but no yield drop, a small yield plateau is found at room temperature. The subsequent strain hardening shows a maximum at 200°C, when serrated flow occurs and the ductility is reduced. The high strain-rate results show a considerable drop of stress at yield. The post-yield flow stress decreases steadily with increasing temperature, throughout the temperature range investigated. At room temperature and below, the strain-hardening rate becomes negative at large strains. The adiabatic temperature rise in the dynamic tests was computed on the assumption that the plastic work is entirely converted to heat. This enabled the isothermal dynamic stress-strain curves to be calculated, and showed that considerable thermal softening took place. The initial response to a strain-rate jump is approximately elastic, and has a magnitude which increases with decrease of testing temperature; it is little affected by the amount of prestrain. At 200 and 400° C, a yield drop occurs after the initial stress increment. The post-jump flow stress is always greater than that for the same strain in a constant-rate dynamic test, the strain-hardening rate becoming negative at large strains or low testing temperature. This observed effect of strain-rate history cannot be explained by the thermal softening accompanying dynamic deformation. These and other results concerning total ductility under various strain-rate and temperature conditions show that strain-rate history strongly affects the mechanical behavior of the mild steel tested and, hence, should be taken into account in the formulation of constitutive equations for that material.     

Chaotic phenomena in the dynamic buckling of an elastic-plastic column under an impact

The present study is concerned with the dynamic anomalous response of an elastic-plastic column struck axially by a massm with an initial velocityv ₀. This simple example is considered in order to clarify the influence of the impact characteristics and the material plastic properties on the dynamic buckling phenomenon and particularly on the final vibration amplitudes of the column when it shakes down to a wholly elastic behaviour. The material is assumed to have a linear strain hardening with a plastic with a plastic reloading allowed. These material properties are the reason a number of elastic-plastic cycles to be realized prior to any wholly elastic stable behaviour, which causes different amounts of energy to be absorbed due to the plastic deformations.The column exhibits two types of behaviour over the range of the impact masses — a quasi-periodic and a chaotic response. The chaotic behaviour is caused by the multiple equilibrium states of the column when any small changes in the loading parameters cause small changes in the plastic strains which result in large changes in the response behaviour. The two types of behaviour are represented by displacement-time and phase-plane diagrams. The sensitivity to the load parameters is illustrated by the calculation of a Lyapunov-like exponent. Poincaré maps are shown for three particular cases.Notation c elastic wave propagation speed - m impact mass - m _c column mass - s step of the spatial discretization - t time - u(x,t) axial displacement - v ₀ initial velocity - w ₀(x) initial imperfections - w(x,t)+w ₀(x) total lateral displacements - x axial axis - z axis along the column thickness - A cross-section areahb - E Young's modulus - E _t hardening modulus (Figure 2) - M(x,t) bending moment - N(x,t) axial force - impact mass ratiom/m _c - (x,z) strain - Lyapunov-like exponent - material density - (x,z) stress     

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.     

Numerical Investigation on the Hydrogen-Assisted Start-Up of Methane-Fueled, Catalytic Microreactors

The hydrogen-assisted start-up of methane-fueled, catalytic microreactors has been investigated numerically in a plane-channel configuration. Transient 2-D simulations have been performed in a platinum-coated microchannel made of either ceramic or metallic walls. Axial heat conduction in the solid wall and surface radiation heat transfer were accounted for. Simulations were performed by varying the inlet pressure, the solid wall thermal conductivity and heat capacity, and comparisons were made between fuel mixtures comprising 100% CH₄ and 90% CH₄?C10% H₂ by volume. A significant reduction in the ignition (t _ig) and steady-state (t _st) times was evident for microreactors fed with hydrogen-containing mixtures in comparison to pure methane-fueled ones, for all pressures and reactor materials investigated, with hydrogen having a direct thermal rather than chemical impact on catalytic microreactor ignition. The positive impact of H₂ addition was attenuated as the pressure (and the associated CH₄ catalytic reactivity) increased. Reactors with low wall thermal conductivity (cordierite material) benefited more from hydrogen addition in the fuel stream and exhibited shorter ignition times compared to higher thermal conductivity ones (FeCr alloy) due to the creation of spatially localized hot spots that promoted catalytic ignition. At the same time, the cordierite material required shorter times to reach steady state. Microreactor emissions were impacted positively by the addition of hydrogen in the fuel stream, with a significant reduction in the cumulative methane emissions and no hydrogen breakthrough. Finally, gas-phase chemistry was found to elongate the steady-state times for both ceramic and metallic materials.     

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.     

Crack-tip fields in steady crack-growth with linear strain-hardening

Singular stress and strain fields are found at the tip of a crack growing steadily and quasi-statically into an elastic-plastic strain-hardening material. The material is characterized byJ₂ flow theory together with a bilinear effective stress-strain curve. The cases of anti-plane shear, plane stress and plane strain are each considered. Numerical results are given for the order of the singularity, details of the stress and strain-rate fields, and the near-tip regions of plastic loading and elastic unloading.