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.     

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 t₀ of time t, and the growth rate at time t₀ 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 t₀ 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.     

Numerical study of shear band instability and effect of cavitation on the response of a specimen under undrained biaxial loading

This paper presents an extension of the local second gradient model to multiphasic materials (solids particles, air, water) and including the cavitation phenomenon. This new development was made in order to model the response of saturated dilatant materials under deviatoric stress and undrained conditions and possibly, in future, the behavior of unsaturated soils. Some experiments have showed the significance of cavitation for the hydromechanical response of materials. However, to date and as far as we are aware, no attempt was made to implement the cavitation as a phase change mechanism with a control of pore pressure. The first part of the results section explores the effects of permeability, dilation angle and loading rate on the stability of shear bands during a localization event. The reasons underlying the band instability are discussed in detail, which helps defining the conditions required to maintain stability and investigating the effects of cavitation without parasite effect of materials parameters or loading rate. The model showed that, if a uniform response is obtained, cavitation triggers localization. However, in case of a localized solution, cavitation follows the formation of the shear band, with the two events being quite distinct.     

Shear band analysis and shear moduli calibration

Strain localization is a well known phenomenon, generally associated with plastic deformation and rupture in solids, especially in geomaterials. In this process, deformation is observed to concentrate in narrow zones called shear bands. This phenomenon has been studied extensively in the last 20 years by different researchers, experimentally, theoretically and numerically. A criterion for the onset of localization can be predicted solely on the basis of the constitutive law of the material, using the so-called shear band analysis. This criterion gives the critical orientation, and the critical stress state and strain for a given loading history. An important point, already stressed by Vardoulakis in 1980, is that in particular, out-of-axes shear moduli play a central role in the criterion. These are the moduli involved in the response to a deviatoric stress increment with principal axes oriented at 45° from total stress principal axes. Out-of-axes shear moduli are difficult parameters to calibrate; common tests, with fixed principal stress and strain directions, do not provide any information on these moduli, as long as they remain homogeneous. Still, real civil engineering and environmental problems are definitely not simple axisymmetric triaxial tests; practical modeling involves complex stress paths, and need complex parameters to be calibrated. Only special tests, like compression–torsion on hollow cylinder tests, or even more complex tests can be used for shear moduli calibration. However, shear band initiation in homogeneous, fixed-axes tests does activate out-of-axes shear. Hence, it is natural that shear band analysis makes shear moduli enter into the analysis.Then, a typical inverse analysis approach can be used here: experimental observation of strain localization in triaxial tests can be used together with a proper shear band analysis for the model considered, in order to determine out-of-axes shear moduli.This approach has been used for a stiff marl in the framework of a calibration study on a set of triaxial tests. The steps of the method are presented, and the bifurcation surface in the stress space is exhibited.     

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.     

On evolution of thermo-plastic shear band

The present paper briefly reviews analytical studies of the evolution of thermoplastic shear band, i.e. emergence from uniform deformation, post-instability growth and late stage behaviour. The case studied is the simple shear of temperature and rate-dependent materials with heat transfer. Uniform mode exists before a critical state, if no heat flows out of testpiece. Upon reaching the critical state, bifurcation appears as a result of disturbances, which leads to instability and the formation of narrow shear band. Initially, the band, due to temperature disturbance, can shrink with increasing temperature and strain rate owing to unsteady flow. Then heat conduction dominates and causes the shear band to expand. The postmortem appearance of thermo-plastic shear band manifests itself as balance of plastic work rate and heat diffusion. Melting may also take place within the band.     

Experimental study of shear flow instability

The results of experiments carried out in order to study the instability of a free shear layer of rotating barotropic fluid at high Reynolds numbers are presented.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.3, pp. 167–173, May–June, 1992.     

Plasticity of damaged solids and shear band localization

Summary The main objective of the paper is the investigation of shear band localization conditions for finite elastic-plastic rate independent deformations of damaged solids. The first part of the paper is devoted to the formulation of the constitutive relations for elastic-plastic solids when isotropic and kinematic hardening effects and the micro-damage process are taken into consideration. The isotropic work-hardening effect is incorporated in the theory directly by defining the work-hardening-softening material function while the kinematic hardening effect and the softening effect generated by the micro-damage process are described by means of the internal state variable method. The second part of the paper aims at the investigation of the localization of plastic deformations. Different effects on the localization phenomenon are investigated. Particular attention is focused on kinematic hardening and micro-damage effects. It has been found that the influence of these two cooperative phenomena on the onset of localization within shear bands has synergetic nature. The results obtained are in good agreement with recent experimental observations.

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Plastizität von geschädigten Feststoffen und Lokalisierung in Scherzonen

Übersicht Hauptgegenstand der Arbeit ist die Untersuchung der Bedingungen, die bei großen elastischplastischen Formänderungen von geschädigten, formänderungsgeschwindigkeitunabhängigen Feststoffen zur Lokalisierung in Scherzonen führen. Der erste Teil dient der Formulierung des Stoffgesetzes für elastischplastische Werkstoffe mit isotroper und kinematischer Verfestigung sowie Mikro-Schädigung. Die isotrope Verfestigung wird unmittelbar durch eine Verfestigungs-Entfestigungsfunktion berücksichtigt, während die kinematische Verfestigung und die Entfestigung infolge Mikro-Schädigung durch innere Zustandsgrößen beschrieben werden. Der zweite Teil befaßt sich mit der Lokalisierung der plastischen Formänderung, wobei verschiedene Einflüsse untersucht werden. Besondere Aufmerksamkeit wird auf die kinematische Verfestigung und Mikro-Schädigung gerichtet. Es stellt sich heraus, daß beide Erscheinungen bei der Lokalisierung in Scherzonen zusammenwirken. Die Ergebnisse stehen in guter Übereinstimmung mit neueren experimentellen Beobachtungen.

     

The mechanical condition of shear band bifurcation

Based on the understanding of the role played by the strain-softening effect in the formation of shear band bifurcation, this paper investigates (a) What is the most favourable condition that stimulates the occurrence of shear band? (b) With what model and characterizing parameters can the curved type of band bifurcation be simulated?     

边坡稳定的剪切带计算

为了解决边坡稳定分析中剪切带有限元网格的依赖性问题,采用梯度塑性理论,从本构关系中引入特征长度入手,建立计算模型。提出了一种8节点缩减积分的梯度塑性单元,并采用梯度塑性理论推导了Drucker-Prager屈服准则的软化模型的有限元格式,在ABAQUS中进行了二次开发,嵌入了本文提出的8节点单元和本构模型,并用ABAQUS软件进行了边坡剪切带的计算。计算结果表明,本文提出的方法消除了经典有限元计算的网格依赖性问题,可以得到与单元剖分无关的稳定的剪切带宽度。本文所提出的方法可适用于其他场合的剪切带计算。     

Adaptive remeshing for shear band localization problems

Summary Failure of earth structures or laboratory specimens of soils is often characterized by the existence of bands or surfaces at which strain localizes. Numerical simulation of shear band development has attracted the attention of many research groups during past years. Here it is proposed that adaptive remeshing techniques can be applied to better simulate strain localization problems in geotechnique. The algorithm has been previously applied to compressible fluid dynamics problems to capture discontinuities such as shocks. A new refinement functional has been introduced to improve quality of produced meshes. Finally, the algorithm is applied to solve inception and development of shear bands on both homogeneous stress fields and non-homogeneous stress fields.

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Adaptive Netzverfeinerung bei Problemen mit Scherfugen-Lokalisierung

Übersicht Das Versagen von Erdbauwerken oder Bodenproben im Labor ist oft durch die Existenz von Flächen mit lokalisierter Deformation gekennzeichnet. Zahlreiche Forschungsgruppen haben sich in den letzten Jahren mit der numerischen Simulation der Scherfugen-Bildung beschäftigt. Zur Verbesserung der Simulation von Scherfugen-Lokalisierung bei bodenmechanischen Problemen wird in diesem Beitrag eine adaptive Netzverfeinerung vorgeschlagen. Dieser Weg wurde schon früher in der Dynamik kompressibler Flüssigkeiten beschritten, um Unstetigkeiten wie Schockfronten zu erfassen. Als Kriterium für die Verfeinerung des Netzes wird in diesem Beitrag ein neues Funktional benutzt. Schließlich wird der Algorithmus auf den Beginn und die Weiterentwicklung von Scherfugen bei homogenen und inhomogenen Spannung zuständen angewandt.

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Presented at the workshop on Numerical Methods for Localization and Bifurcation of Granular Bodies, held at the Technical University of Gdansk (Poland), September 25–30, 1989     

A shear flow instability in cylindrical geometry

Conclusion The results show, that this model experiment is useful for studying the interaction between two parallel shear layers in a rotating system. The structures of the observed instabilities exhibit some similarities with the wake flow behind cylindrical obstacles. Additional investigations in analyzing the dynamical behaviour of the shear instabilities are necessary to get a deeper insight into the physical mechanism. Up to that it is an open question, whether the physical effects described by Meiburg (1987), play an important role in these shear layer instabilities.     

The effect of stress induced anisotropy on shear band formation

Summary This paper describes the effect of non-coaxiality arising from material anisotropy on bifurcation phenomena such as shear band formation. The elasto-plastic model originally proposed by Sekiguchi and Ohta [7] is one of the typical models which include anisotropy and it is used to examine the effect of anisotropy on shear band formation. First, we shall extend this elasto-plastic model for infinitesimal strain to a model for finite strain and discuss the mathematical structure of this model. The stress induced anisotropy is found to bring about a vertex-like effect, such as the non-coaxiality between the Cauchy stress tensor and a plastic stretching tensor, into the constitutive relation. Then, we shall examine the effect of this non-coaxiality on bifurcation conditions in relation to the material rigidity which changes with the angle of simple shear. Finally, it will be concluded that this non-coaxiality arising from the anisotropy does not contribute much to triggering instability by localization of the deformations which result in shear band formation, while on the other hand, the non-coaxiality due to the yield vertex effect is rather inclined towards instability by localization of the deformations.

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Der Einfluß spannungsinduzierter Anisotropie auf die Scherzonenbildung

Übersicht Beschrieben wird der Einfluß anisotropie-bedingt verschiedener Achsen von Spannung und Deformationsgeschwindigkeit auf Verzweigungsphänomene wie die Scherzonenbildung. Das elastisch-plastische Modell von Sekiguchi und Ohta wird als typisch anisotropes Modell zur Untersuchung des Einflusses der Anisotropie auf die Scherzonenbildung benutzt. Zunächst wird es zu einem Modell für endliche Deformationen erweitert und seine mathematische Struktur erörtert. Es zeigt sich, daß die spannungsinduzierte Anisotropie einen Effekt ähnlich einer Fließortspitze erzeugt wie z. B. verschiedene Achsen von Cauchy-Spannungstensor und plastischem Strecktensor. Danach wird in Zusammenhang mit der Materialsteifigkeit, die vom Winkel der einfachen Scherung abhängt, der Einfluß verschiedener Achsen auf Verzweigungsbedingungen untersucht. Es wird gefolgert, daß anisotropie-bedingt verschiedene Achsen wenig zur Auslösung der Instabilität durch lokalisierte Deformation beitragen, andererseits verschiedene Achsen infolge einer Fließortspitze die Neigung zur lokalisierten Deformation unterstützen.

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Presented at the workshop on Numerical Methods for Localization and Bifurcation of Granular Bodies, held at the Technical University of Gadansk (Poland), September 25–30, 1989     

Flow instability and shear localization in a drilling mud

To have a better knowledge of problems occurring with drilling fluids in complex wells, we carried out a detailed rheological analysis of a typical drilling mud at low shear rates using both conventional rheometry and MRI velocimetry. We show the existence of a viscosity bifurcation effect: Below a critical stress value, the mud tends to completely stop flowing, whereas beyond this critical stress, it reaches an apparent shear rate larger than a finite (critical) value, and no stable flows can be obtained between this critical shear rate value and zero. These results are confirmed by MRI velocity profiles, which exhibit a slope break at the interface between the solid and the liquid phases inside the Couette geometry. Moreover, this viscosity bifurcation is a transient phenomenon, the progressive development of which can be observed by MRI. A further examination of MRI data shows that, in the transient regime, the shear rate does not vary monotonously in the rheometer gap and is particularly large along the outer (rough) cylinder, which might be at the origin of the development of a region of constant shear rate in the apparent flow curve.     

On criteria for dynamic adiabatic shear band propagation

In this work, we postulate the physical criterion for dynamic shear band propagation, and based on this assumption, we implement a numerical algorithm and a computation criterion to simulate initiation and propagation of dynamic adiabatic shear bands (ASBs). The physical criterion is based on the hypothesis that material inside the shear band region undergoes a dynamic recrystallization process during deformation under high temperature and high strain-rate conditions. In addition to providing a new perspective to the physics of the adiabatic shearbanding process and identifying material properties that play a crucial role in defining the material's susceptibility to ASBs, the proposed criterion is instrumental in numerical simulations of the propagation of ASBs when multi-physics models are adopted to describe and predict the complex constitutive behavior of ASBs in ductile materials. Systematic and large scale meshfree simulations have been conducted to test and validate the proposed criterion by examining the formation, propagation, and post-bifurcation behaviors of ASBs in two materials, 4340 steel and OFHC copper. The effects of heat conduction, in particular the length scale introduced by heat conduction, are also studied. The results of the numerical simulations are compared with experimental observations and a close agreement is found for various characteristic features of ASBs, such as the shear band width, speed of propagation, and maximum temperature.     

Analysis of shear band instabilities in sintered metals

For structural components made of sintered metals the porosity may be rather high, and this affects conditions for material failure. A simple model for analysis of shear band development in uniformly strained solids is used here to study the effect of rather high initial porosities on the onset of a material instability. The elastic plastic behaviour of the material is represented by a material model, which combines the Gurson model, relevant to rather low porosities, with the FKM model, developed for high porosity powder compacts. Predictions are shown for various levels of initial porosity and for various levels of initial material imperfections, considering both plane strain and axisymmetric conditions outside the shear band. Also a comparison of localization predictions by the Gurson model, the FKM model and the combined model is presented.     

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.