Shear Band Spacing Research Articles

Structure of Shear Bands in Zirconium-Based Metallic Glasses Observed by Transmission Electron Microscopy

ABSTRACTWe have used transmission electron microscopy (TEM) to investigate the structure of shear bands produced by bending electron-transparent Zr52.5Cu17.9Ni14.6Al10Ti5metallic glass specimens. Shear bands were located by comparing the structure of the specimens before and after deformation. The shear band spacing is influenced by the structure of the specimen; portions of the specimen with a significant population of nanocrystals show a smaller separation between shear bands. Quantitative high resolution TEM analysis based on ratio technique has been used to explore the defect structure in shear bands. High density and void-like defects with size of about 1 nm were found in shear bands formed in both amorphous and nanocrystalline areas. A simple model was proposed to explain the formation of these defects.

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. # 2001 Elsevier Science Ltd. All rights reserved.

Improved mechanical behavior of bulk metallic glasses containing in situ formed ductile phase dendrite dispersions

In this paper, X-ray diffraction, microstructural, and mechanical property results are presented for a ductile metal reinforced bulk metallic glass (BMG) matrix composite based on Zr–Ti–Cu–Ni–Be bulk glass forming compositions. Solute partitioning in the molten state yields an equilibrium microstructure consisting of ductile Ti–Zr–Nb β-phase primary dendrites, with the b.c.c. structure, in a Zr–Ti–Nb–Cu–Ni–Be BMG matrix. Under mechanical loading highly organized shear band patterns are formed throughout the sample, where the shear band spacing is coherent with the periodicity of the dendrites. This dramatically increases the plastic strain to failure, impact resistance, and toughness of the material. These results are the first to show how microstructural inhomogeneities can be used to control the initiation and propagation of localized shear bands in metallic glasses under a variety of unconstrained loading conditions.

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.

Self-organization of shear bands in Ti, Ti-6%Al-4%V, and 304 stainless steel

Adiabatic shear bands self-organize themselves: this phenomenon was investigated in three different alloys ( stainless steel 304L, Ti, and Ti-6%Al-%V) through the radial collapse of a thick-walled cylinder under high-strain-rate deformation (∼10 4 s -1 ). This method was used to examine the shear-band initiation, propagation, as well as spatial distribution. The spacing of shear bands in the three materials varied widely and showed considerable differences during the initiation and propagation. Shear-band spacing is compared with the theories proposed by Grady and Kipp (momentum diffusion), Wright and Ockendon (perturbation), and Molinari (perturbation plus work hardening). Additionally, the effect of microstructural inhomogeneities on initiation is discussed. A discontinuous growth mode for shear localization under periodic perturbation is proposed based on two-dimensional growth considerations, not incorporated into the one-dimensional perturbation and momentum diffusion theories.

Effect of material parameters on shear band spacing in work-hardening gradient dependent thermoviscoplastic materials

We study thermomechanical deformations of a viscoplastic body deformed in simple shear. The effect of material elasticity is neglected but that of work hardening, strain-rate hardening, thermal softening, and strain-rate gradients is considered. The consideration of strain-rate gradients introduces a material characteristic length into the problem. A homogeneous solution of the governing equations is perturbed at different values t 0 of time t, and the growth rate at time t 0 of perturbations of different wavelengths is computed. Following Wright and Ockendon's postulate that the wavelength of the dominant instability mode with the maximum growth rate at time t 0 determines the minimum spacing between shear bands, the shear band spacing is computed. It is found that for the shear band spacing to be positive, either the thermal conductivity or the material characteristic length must be positive. Approximate analytical expressions for locally adiabatic deformations of dipolar (strain-rate gradient-dependent) materials indicate that the shear band spacing is proportional to the square-root of the material charateristic length, and the fourth root of the strain-rate hardening exponent. The shear band spacing increases with an increase in the strain hardening exponent and the thermal conductivity of the material.

Shear band spacing in gradient-dependent thermoviscoplastic materials

We study thermomechanical deformations of a viscoplastic body deformed in simple shear. The strain gradients are taken as independent kinematic variables and the corresponding higher order stresses are included in the balance laws, and the equation for the yield surface. Three different functional relationships, the power law, and those proposed by Wright and Batra, and Johnson and Cook are used to relate the effective strain rate to the effective stress and temperature. Effects of strain hardening of the material and elastic deformations are neglected. The homogeneous solution of the problem is perturbed and the stability of the problem linear in the perturbation variables is studied. Following Wright and Ockendon's postulate that the wavelength whose initial growth rate is maximum determines the minimum spacing between adjacent shear bands, the shear band spacing is computed. It is found that the minimum shear band spacing is very sensitive to the thermal softening coefficient/exponent, the material characteristic length and the nominal strain-rate. Approximate analytical expressions for the critical wave length for heat conducting nonpolar materials and locally adiabatic deformations of gradient dependent materials are also derived.

Self-organization in the initiation of adiabatic shear bands

The radial collapse of a thick-walled cylinder under high-strain-rate deformation (∼104s−1) was used for the investigation of shear-band initiation and pattern development in titatium. Experiments were carried out in which the collapse was arrested in two stages, in order to observe the initiation and propagation of shear bands. The occurrence of shear bands to accommodate plastic deformation in response to external tractions is a collective phenomenon, because their development is interconnected. The bands were observed to form on spiral trajectories and were periodically spaced. The spacing of the shear bands decreased with the progression of collapse, and was equal to approximately 0.6mm in the final stage of collapse. The shear-band spacing was calculated from two existing models, based on a perturbation analysis and on momentum diffusion. Values of 0.52 and 3.3mm were obtained with material parameters from quasi-static and dynamic experiments. The predictions are found to give a reasonable first estimate for the actual spacings. The detailed characterization of the shear-band front leads to an assessment of the softening mechanisms inside a shear localization region. The initiation of localization takes place at favorably oriented grains and becomes gradually a continuous process, leading eventually to dynamic recrystallization.

Shear localization and chemical reaction in Ti–Si and Nb–Si powder mixtures: Thermochemical analysis

Chemical reactions in Ti–Si and Nb–Si powder mixtures were initiated in regions of high plastic strain induced by high-strain-rate deformation. These regions of high localized plastic strain had thicknesses of 10–25 μm and a characteristic spacing of 600–1000 μm. Scaling up of the experiments revealed a shear-band spacing that is constant and dictated by material and deformation parameters. The generation of heat due to plastic deformation and chemical reaction is treated in a one-dimensional calculation. The calculations are in qualitative agreement with experimental results: shear bands can serve as ignition regions for the propagation of the reaction throughout the entire specimen for Ti–Si, whereas in the Nb–Si system (that has a much lower enthalpy of reaction), the reaction is always localized in the shear bands. These results enable the estimation of a reaction time in the Ti–Si mixture (∼10 ms) and of a critical global strain required for the complete reaction to take place (εeff=0.38). This is a new regime of reaction, intermediate between combustion synthesis and shock compression synthesis.

Shear localization and chemical reaction in high-strain, high-strain-rate deformation of Ti–Si powder mixtures

Ti–Si mixtures were subjected to high-strain-rate deformation at a pressure below the threshold for shock-wave initiation. Whereas the collapse of interparticle pores did not initiate reaction, regions of localized macro-deformation initiated reaction inside shear bands at sufficiently high strains ( γ∼10), and propagation of the reaction through the entire specimen at higher strains ( γ∼20–40). This study demonstrates that temperature increases in shear localization regions can initiate chemical reaction inside a reactive powder mixture. The shear band spacing was ∼0.6–1 mm. Thermodynamic and kinetic calculations yield the reaction rate outside the shear bands, in the homogeneously deformed material, which has a maximum value of 20 s −1 at 1685 K.

Collective behavior and spacing of adiabatic shear bands

The failure of metals subjected to high strain rates is frequently related to the collective development of adiabatic shear bands which results in a patterning depending on the material properties and the loading conditions. In this paper, the spacing between adiabatic shear bands is characterized by analytical means in a one-dimensional formulation. Using a perturbation analysis, a dominant instability mode can be characterized, whose wavelength is related to the shear-band spacing. Explicit solutions are found for materials with no strain hardening. Asymptotic developments are used to obtain results that account for strain hardening. Comparisons are made with experimental results and other existing models.

Scaling laws for adiabatic shear bands

Mathematical analysis of a simple, one-dimensional, canonical problem has yielded a number of scaling laws that relate various characteristic features of an adiabatic shear band to the physical properties of the material and the ambient conditions. Specific formulas have been obtained from solutions to linear and non-linear problems coupled with asymptotic representations of the results. Examples are shear band width, most sensitive strain rate and shear band spacing. Other results show the equivalence of initial fluctuations in strength or temperature, and still others show how to scale the defect in formulas for estimating the timing of localization. The paper summarizes previous results and presents some new formulas that show how strain rate sensitivity affects band spacing, number density and morphology as a function of strain rate.

Properties of functionally graded NiAl/Al 2O 3 composites : Miller, D.P., Lannutti, J.J., Soboyejo, W.O. and Noebe, R.D. Proc. Conf. Structural Intermetallics, Champion, PA, USA, 26–30 Sept. 1993, pp. 783–790

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.

A thermal-transient test of an FBR piping-bellows model: Saito, T., Umeda, H., Kanazawa, S. Watashi, K. and Imazu, A. Int. J. Pressure Vess. Piping 1990 44, (1), 117–135

Four-point-bend fatigue experiments are conducted on rectangular beam samples of a Zr50Cu37Al10Pd3 (in atomic percent) bulk metallic glass (BMG). In general, a fatigue crack initiates from the corner of the beam sample at the tensile stress surface and subsequently grows inside during cyclic loading. As a result, many shear bands form around the crack growth path due to the plastic deformation at the crack tip. Moreover, the size of the shear-band zone shows a good relationship with the stress intensity factor range, which is similar to the plastic zone in the crystalline alloys. Moreover, the shear-band spacing is consistent with the coarse fatigue striation spacing, which implies that the fatigue-crack growth in BMGs is related to the formation of shear bands. A reasonable mechanism is proposed to understand the fatigue-crack-growth behavior in BMGs.

A variational principle for gradient plasticity

We elaborate on a generalized plasticity model which belongs to the class of gradient models suggested earlier by Aifantis and co-workers. The generalization of the conventional theory of plasticity has been accomplished by the inclusion of higher-order spatial gradients of the equivalent plastic strain in the yield condition. First it is shown how these gradients affect the critical condition for the onset of localization and allow for a wavelength selection analysis leading to estimates for the width and or spacing of shear bands. Due to the presence of higher-order gradients, additional boundary conditions for the equivalent plastic strain are required. This question and also the associated problem of the formulation and solution of general boundary value problems were left open in the previous work. We demonstrate here that upon assuming a certain type of additional boundary conditions, the structural symmetries of the gradient-dependent constitutive model are such that there exists a variational principle for the displacement rates and the rate of the equivalent plastic strain. The variational principle can serve as a basis for the numerical solution of boundary value problems in the sense of the finite element method. Explicit expressions for the tangent stillness matrix and the generalized nodal point forces are given.

A Gradient-Dependent Flow Theory of Plasticity: Application to Metal and Soil Instabilities

We propose a gradient-dependent flow theory of plasticity for metals and granular soils and apply it to the problems of shear banding and liquefaction. We incorporate higher order strain gradients either into the constitutive equation for the flow stress or into the dilantancy condition. We examine the effect of these gradients on the onset of instabilities in the form of shear banding in metals or shear banding and liquefaction in soils under both quasi-static and dynamic conditions. It is shown that the higher order gradients affect the critical conditions and allow for a wavelength selection analysis leading to estimates for the width or spacing of shear bands and liquefying strips. Finally, a nonlinear analysis is given for the evolution of shear bands in soils deformed in the post-localization regime.

The growth of unstable thermoplastic shear with application to steady-wave shock compression in solids*

The catastrophic growth of unstable thermoplastic shear following the transition from homogeneous deformation to heterogeneous localized deformation through distributed shear banding is studied through approximate analytic and computational methods. The calculations provide expressions for shear band widths, spacing, catastrophic growth times and the rate of stress communication between shear bands. The optimum shear band width and spacing are found to be consistent with a minimum work principle. The model predicts that the product of the energy dissipated and the localization time in the shear localization process is invariant with respect to changes in the driving strain rate. Such behavior has been noted in the steady-wave shock compression of a number of solids. The calculations are applied to heterogeneous shear localization observed in the shock compression of aluminum.

Local inertial effects in dynamic fragmentation

A general definition of dynamic fragmentation can encompass any impulsive process which partitions a body of material into discrete domains. Two examples are fragmentation due to brittle fracture under impact loading and fragmentation due to shear banding in shock-compression plastic deformation. In application, prediction of fragment size or shear band spacing is frequently either the objective, or else requisite to understanding the process. An approach is presented whereby surface or interface area created in the fragmentation process is governed by an equilibrium balance of the surface or interface energy and a local inertial or kinetic energy. Fragment size can be approximately related to surface or interface area. Relations provided by the analysis compare well with experimental dynamic fracture and shock-wave shear-band results.