A new constitutive model for shear banding instability in metallic glass

Inelastic deformation of metallic glass is through shear banding, characterized by significantly localized deformation and emerged expeditiously under certain stress state. This study establishes a new constitutive model addressing the physical origin of the shear banding. In the modeling, the atomic structural change and the free volume generation are embodied by the plastic shear strain and the associated dilatation. The rugged free energy landscape is adopted to naturally reflect the rate-independent flow stress and flow serrations. Based on this, the conditions for the onset of shear banding instability are established, which enables the explicit calculation of the shear band inclination angle and its extension speed. The study concludes that shear band angle is significantly influenced by the diltancy factor and pressure sensitivity, that a shear band does not increase its thickness once emanated from a deformation unit, that the spreading speed of a shear band is intersonic, and that more shear bands, which lead to higher ductility, can be induced by high strain rates or by the introduction of a second material phase. The analysis also demonstrates that the ductility of metallic glass depends on the sample geometry and/or the stress state.     

影响金属玻璃中剪切带行为的微观机制

论文对金属玻璃发生剪切失稳形成剪切带的行为进行了分析,得到了其发生剪切失稳时的临界自由体积浓度,预测结果与实验观察和模拟结果吻合;利用两种方法对其剪切带厚度进行了预测,结果表明基于剪切失稳临界波长预测金属玻璃剪切带厚度的方法只在发生剪切失稳后极短的时间内有效,对成熟剪切带厚度的预测必须考虑自由体积的扩散效应;考察了金属玻璃的宏微观材料参数对其剪切带厚度的影响及其微观机制,发现金属玻璃剪切带厚度对其宏观材料参数(泊松比)不敏感,对与剪切相变区相关的微结构参数敏感.     

Numerical simulation of shear band formation in a frictional-dilational material

Summary The explicit, finite difference code FLAC is used to model shear band development in isotropic, elastic-plastic, Coulomb, non-associated, non-hardening materials. The code reproduces shear band inclinations which depend on both the angle of friction and the angle of dilation. Localization occurs at the yield point. Shear band width is sensitive to the size of the finite difference mesh but is controlled also by the magnitudes of the friction and dilation angles. The distribution of shear bands throughout the specimen is also influenced by the values of friction and, to a lesser extent, of dilation. There is no suggestion that numerical instability or numerical truncation errors are responsible for shear band nucleation.

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Numerische Simulation von Scherfugenbildungen in druckabhängigen, dilatanten Materialien

Übersicht In numerisch simulierten Experimenten an einem gleichförmig belasteten rechteckigen Block werden die Möglichkeiten des Finite-Differenzen-Programms FLAC zur Darstellung von Scherfugenbildungen überprüft. Wir betrachten elastisch-ideal plastische Materialien vom Coulomb-Typ mit nicht-assoziierter Fließregel. Es ergeben sich Scherfugenorientierungen, die sowohl vom Wert des Reibungswinkels als auch vom Wert des Dilatanzwinkels abhängen. Die Scherfugendicken hängen sowohl von der Maschenweite des Differenzennetzes als auch vom Reibungs- und Dilatanzwinkel ab. Der Wert des Reibungswinkels und in geringem Maße auch der Wert des Dilatanzwinkels beeinflussen die Art der Scherfugenverteilung innerhalb des Blocks. Die Ergebnisse der Berechnungen werden, soweit möglich, mit analytischen Lösungen verglichen.

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Presented at the workshop on Limit Analysis and Bifurcation Theory, held at the University of Karlsruhe (FRG), February 22–25, 1988     

Shear stability of metallic glasses

The effect of sample size on the shear deformation and compressive plasticity of different metallic glasses were investigated. The experimental results showed that the deformation and fracture behaviors of samples prepared from chemically identical Zr-, Ti-, Fe-, or Mg-based metallic glass ingots were strongly dependent on the sample size and machine stiffness, and a super-high compressive plasticity was achieved in the Zr-based metallic glasses with sample size of 1.0 mm in width. It is also found that the sample size can significantly influence the density of elastic energy dissipated in the shear band: with sample size decreasing and machine stiffness increasing, the density of the elastic energy dissipated in the shear band of metallic glasses is prominently decreased, thus the shear deformation turns to be more stable, resulting in the improvement of plasticity in ductile metallic glasses and the transition from fragmentation fracture to shear fracture in brittle metallic glasses. This finding is important for the potential applications of the present metallic glasses and for designing new metallic glasses with better mechanical properties.     

剪切带状分叉

本文从扩散型解法出发，结合实验，揭示了剪切带状分叉的主要特征；进而，论述材料分叉与结构屈曲（分叉）的不同之处。文中结果表明：在诱发剪切带过程中，材料内空洞的激化速率是控制因素而不在于空洞体积比的自身数量。     

Cone resistance in sand by incremental method

A new method has been developed for the determination of cone resistance under drained conditions. Numerical methods are used for the solution of the differential equations of plasticity theory for soils and for the determination of the stress states in the soil produced by the penetration of the cone. It is assumed that the stresses produced by the penetration of the cone remain ‘locked in’ the soil and constitute boundary conditions for further penetration. The computation starts with the cone base at the surface and is continued by successively incrementing the depth by a small amount. Charts are given for the computation of cone resistance in sands for various friction angles. The importance of the effect of the shear stresses generated at the surface of the cone and characterized by the interface friction angle, δ, is discussed in detail.     

Modelling of adiabatic shear band development from small imperfections

A model of shear banding is presented which shows how a wide shear band develops from a narrow imperfection in an elasto-viscoplastic material subjected to dynamic shear strain. The model predicts that the width of the shear band is (i) independent of the properties of the initial imperfection and (ii) dependent upon thermal conductivity and strain rate. The dependence upon strain rate is verified qualitatively and quantitatively from experimental results. Finally, the model predicts narrowing of the region of rapid straining with ongoing deformation as is observed in experiment.     

Grain-scale processes governing shear band initiation and evolution in sands

A variety of experimental techniques have been used to advance understanding of strain localization phenomena in sands. However, all of these methods have fallen short in characterizing the evolution of the grain-scale processes that necessarily control shear band formation and growth in sands. This paper presents results of application of the non-destructive displacement measurement technique of digital image correlation (DIC) to measure two- and three-dimensional surface displacements on plane strain and axisymmetric sand specimens over short time steps. The abundance of local displacement data, high level of accuracy, and nearly continuous (spatially and temporally) record of displacement evolution afforded by the DIC technique has finally enabled a means to quantify local displacements to particulate-scale intensity. The data have been used to evaluate the local displacement mechanisms leading to the triggering of the formation of persistent shear bands, the timing of shear band formation with regard to the achievement of peak stress, and the character of displacements within fully formed shear bands. Insights are offered regarding the relation between strain localization and global stress-strain behavior, and the ensuing interpretations of shear banding as a hardening or softening phenomenon. Comparison of behavior between plane strain and triaxial tests offer additional perspective on the influences of three-dimensional stresses and boundary conditions on shear banding. The results further shed light on the micro-deformation mechanisms (i.e. buckling columns) responsible for the observed local strain non-uniformities that characterize “steady-state” shear band evolution.     

Kink band instability in layered structures

A recent two-dimensional prototype model for the initiation of kink banding in compressed layered structures is extended to embrace the two propagation mechanisms of band broadening and band progression. As well as interlayer friction, overburden pressure and layer bending energy, the characteristics of transverse layer compressibility and foundation stiffness are now included. Experiments on constrained layers of paper show good agreement with the predictions of angle of orientation, kink band width and post-kink load-deflection response obtained from the model.     

岩土介质中局部化变形研究的一些新进展

岩土中的剪切带产生通常是断裂和失稳的前兆, 常引起滑坡等地 质灾害, 对油气或水合物等矿藏的开采容易导致压实带形成. 压实带的产生使岩土孔隙减小, 颗粒破碎, 从而导致沉陷、对油气的挤压等.本文重点对剪切带的宽度和压实带的研究现状作综述, 阐述在该问题的研究上从微观 观测到宏观表述的重要性. 通过分析, 认为还有如下问题值得研究: 怎样 准确得出局部化变形带的宽度和方向?局部化变形带形成后孔隙水和油气的渗流会发生什么 变化?颗粒层次的微结构特性是通过什么方式影响宏观表现的等等.     

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.     

Adiabatic shear banding in high speed machining of Ti–6Al–4V: experiments and modeling

An experimental analysis of orthogonal cutting of a Ti–6Al–4V alloy is proposed. Cutting speeds are explored in a range from 0.01 to 73 m/s by using an universal high-speed testing machine and a ballistic set-up. The evolution of the cutting force in terms of the cutting speed and the development of adiabatic shear banding are analyzed. The shear band width and the distance between bands have been determined by micrographic observations. Their dependence upon the cutting velocity is analyzed. A modeling is proposed which restitutes well this velocity dependence.     

Development of a two-dimensional model of a compliant non-pneumatic tire

An analytical model for a compliant non-pneumatic tire on frictionless, rigid ground is presented. The tire model consists of a thin flexible annular band and spokes that connect the band to a rigid hub. The annular band is modeled using curved beam theory that takes into account deformations due to bending, shearing and circumferential extension. The effect of the spokes, which are distributed continuously in the model and act as linear springs, is accounted for only in tension, which introduces a nonlinear response. The quasi-static, two-dimensional analysis focuses on how the contact patch, vertical tire stiffness and rolling resistance are affected by the stiffness properties of the band and the spokes. A Fourier series representation of the shear strain in the annular band and the complex modulus of the material were used to predict rolling resistance due to steady state rolling. From the analysis point of view, when the wheel is loaded at its hub, the following three distinct regions develop: (1) a support region where the hub hangs by the spokes from the upper part of the flexible band, (2) a free surface region where the spokes buckle and have no effect, and (3) a contact region where the flexible band is supported by the ground without the effect of the spokes. The angular bounds of these three regions are determined by the spoke angle and the contact angle, which are respectively the angle at which the spokes start to engage in tension and the angle that defines the edge of contact. Closed-form expressions of contact stress, stress-resultants and displacements at the centroids of the cross-sections of the flexible band are expressed in terms of these angles, which must be determined numerically. A thorough parametric analysis of quantities of interest for the tire is presented, which can be used to help support the optimal and rational design of compliant non-pneumatic tires. The model was validated by comparison with two computational models using the commercial finite element software ABAQUS and by experimental rolling resistance data.     

An experimental study on spontaneous adiabatic shear band formation in electro-magnetically collapsing cylinders

The formation of shear bands in collapsing thick-walled cylinders (TWC) occurs in a spontaneous manner. The advantage of studying spontaneous, as opposed to forced, shear localization, is that it highlights the inherent susceptibility of the material to adiabatic shear banding without prescribed geometrical constraints. In the case of spontaneous shear localization, the role of microstructure (grain size and grain boundaries) on localization, is still unresolved. Using an electro-magnetic set-up, for the collapse of thick-walled cylinders, we examined the shear band formation and evolution in seven metallic alloys, with a wide range of strength and failure properties. To assess microstructural effects, we conducted systematic tests on copper and Ti6Al4V with different grain sizes. Our results match quite well with previously reported data on much larger specimens, showing the absence of a size effect, on adiabatic shearing. However, the measured shear band spacings, in this study, do not match the predictions of, existing analytical models, indicating that the physics of the problem needs to be better modeled.     

Transient rolling friction model for discrete element simulations of sphere assemblies

The rolling resistance between a pair of contacting particles can be modeled with two mechanisms. The first mechanism, already widely addressed in the DEM literature, involves a contact moment between the particles. The second mechanism involves a reduction of the tangential contact force, but without a contact moment. This type of rotational resistance, termed creep-friction, is the subject of the paper. Within the creep-friction literature, the term “creep” does not mean a viscous mechanism, but rather connotes a slight slip that accompanies rolling. Two extremes of particle motions bound the range of creep-friction behaviors: a pure tangential translation is modeled as a Cattaneo–Mindlin interaction, whereas prolonged steady-state rolling corresponds to the traditional wheel–rail problem described by Carter, Poritsky, and others. DEM simulations, however, are dominated by the transient creep-friction rolling conditions that lie between these two extremes. A simplified model is proposed for the three-dimensional transient creep-friction rolling of two spheres. The model is an extension of the work of Dahlberg and Alfredsson, who studied the two-dimensional interactions of disks. The proposed model is applied to two different systems: a pair of spheres and a large dense assembly of spheres. Although creep-friction can reduce the tangential contact force that would otherwise be predicted with Cattaneo–Mindlin theory, a significant force reduction occurs only when the rate of rolling is much greater than the rate of translational sliding and only after a sustained period of rolling. When applied to the deviatoric loading of an assembly of spheres, the proposed creep-friction model has minimal effect on macroscopic strength or stiffness. At the micro-scale of individual contacts, creep-friction does have a modest influence on the incremental contact behavior, although the aggregate effect on the assembly's behavior is minimal.     

Distinguishing shear banding from shear thinning in flows with a shear stress gradient

Shear banding occurs in complex fluids that exhibit a non-monotonic constitutive instability, such as wormlike micelles, and potentially also in polymeric fluids with presumably monotonic constitutive behavior. However, velocity profiles for shear thinning fluids in geometries possessing a stress gradient, such as Taylor-Couette flow, could be misidentified as shear banding. To address this, we present a model-free experimental procedure to distinguish shear banding from strong shear thinning using high-resolution velocimetry. The approach is developed and validated using simulations using the d-Giesekus model and is based upon the behavior of the width of the apparent interface between the high and low shear rate regions. It is then tested using experimental data for model wormlike micellar solutions. The method allows shear banding to be distinguished from shear thinning in cases where this difference is otherwise indistinguishable. As a by-product, it also provides an estimate of the stress diffusivities for shear banding fluids.     

摩擦与滚阻对被动行走器步态影响的研究

本文研究了圆弧足被动行走器支撑足与地面间的摩擦系数和滚阻系数对被动行走器步态的影响. 首先分别利用扩展的 赫兹接触力模型和LuGre摩擦模型描述了支撑足与地面接触点处的法向支撑力和切向摩擦力,并考虑了行走过程中支撑足 所受的滚动摩阻;其次利用第二类Lagrange方程推导出了该系统的动力学方程,并通过与已有成果的对比确定 了合适的LuGre摩擦模型参数;最后仿真分析了摩擦系数和滚阻系数对被动行走器步态的影响. 研究发现:摩擦系数的改变 虽然对被动行走器行走的平均速度、步幅,以及支撑足接触点处的最大法向接触力的影响较小,但摩擦系数的减小 会改变其行走步态类型,如发生倍周期分岔甚至混沌现象;然而,滚阻系数的改变会对行走器行走的 平均速度、步幅,以及支撑足接触点处的最大法向接触力的影响较大,尚未发现滚阻系数的改变会引起其行走步态的变化.      

Mode II fracture behavior of a Zr-based bulk metallic glass

Shear band formation and fracture are characterized during mode II loading of a Zr-based bulk metallic glass. The measured mode II fracture toughness, K_IIc=75±4 MPa√m, exceeds the reported mode I fracture toughness by ∼4 times, suggesting that normal or mean stresses play a significant role in the deformation process at the crack tip. This effect is explained in light of a mean stress modified free volume model for shear localization in metallic glasses. Thermal imaging of deformation at the mode II crack tip further reveals that shear bands initiate, arrest, and reactivate along the same path, indicating that flow in the shear band leads to permanent changes in the glass structure that retain a memory of the shear band path. The measured temperature increase within the shear band is a fraction of a degree. However, heat dissipation models indicate that the temperature could have exceeded the glass transition temperature for less than 1 ms immediately after the shear band formed. It is shown that this time scale is sufficient for mechanical relaxation slightly above the glass transition temperature.     

Shear banding in strain hardening polycrystals during rolling

Initiation and development of shear band (SB) in f.c.c. strain hardening polycrystals during rolling are modelled in terms of crystallographic texture. The constitutive law of the material is expressed in terms of the texture-dependent normalized yield surface and the critical shear stress which evolves with strain. The normalized yield surface is predicted by the Taylor model as a function of rolling texture. It is shown that a rounded vertex (RV) develops at the loading point as the rolling texture becomes more and more marked. A detailed characterization of the RV is carried out. It is found that the normalized curvature radius of the RV decreases from unity towards zero at very large strain. This allows for a small stress perturbation to induce a shear strain perturbation with a large orientation deviation of deformation. By linearized stability analysis, the condition for initiation of SB from the shear strain perturbation is obtained. Development of SB is analysed by solving the established governing equations of shear banding. It is shown that the conditions for SB initiation and saturation of shear localisation depend strongly on the texture. Based on this model problem, a long discussion is carried out which allows a better understanding of the basic physical origin and saturation of SB in strain-hardening polycrystals, as well as the effects of yield surface curvature and yield surface rotation whose general form is derived.