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.     

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.     

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.     

Shear band propagation from a crack tip subjected to Mode II shear wave loading

Results are reported for pressure–shear plate impact experiments in which pre-cracked 4340 steel plates are subjected to Mode II loading. Experiments show the propagation of a shear band ahead of the initial crack. Finite element simulations are used to interpret the results. Normal and transverse velocity–time profiles measured at the rear surface of the target can be simulated reasonably well using even an elastic model for the material response. A propagating shear band is obtained when the material is modeled as having reduced shearing resistance described by a thermo-viscoplastic power law, and complete loss of shearing resistance when the shear strain reaches a critical value. However, the predicted speed of propagation of the tip of the shear band is substantially less than required to explain the lengths of the bands observed in the experiments. Adjustments of parameters of the power-law model have little effect on the overall length of the band. Possible reasons for differences between predicted and measured shear band speeds are examined. Further reduction in the shearing resistance in the shear band appears to be essential for the simulated bands to be as long as those observed in the experiments.     

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.     

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 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.     

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.     

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。     

Shear banding analysis of geomaterials by strain gradient enhanced damage model

Shear band localization is investigated by a strain-gradient-enhanced damage model for quasi-brittle geomaterials. This model introduces the strain gradients and their higher-order conjugate stresses into the framework of continuum damage mechanics. The influence of the strain gradients on the constitutive behaviour is taken into account through a generalized damage evolutionary law. A weak-form variational principle is employed to address the additional boundary conditions introduced by the incorporation of the strain gradients and the conjugate higher-order stresses. Damage localization under simple shear condition is analytically investigated by using the theory of discontinuous bifurcation and the concept of the second-order characteristic surface. Analytical solutions for the distributions of strain rates and strain gradient rates, as well as the band width of localised damage are found. Numerical analysis demonstrates the shear band width is proportionally related to the internal length scale through a coefficient function of Poisson’s ratio and a parameter representing the shape of uniaxial stress–strain curve. It is also shown that the obtained distributions of strains and strain gradients are well in accordance with the underlying assumptions for the second-order discontinuous shear band boundary and the weak discontinuous bifurcation theory.     

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.     

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.     

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.     

Micromechanics based second gradient continuum theory for shear band modeling in cohesive granular materials following damage elasticity

Gradient theories, as a regularized continuum mechanics approach, have found wide applications for modeling strain localization failure process. This paper presents a second gradient stress–strain damage elasticity theory based upon the method of virtual power. The theory considers the strain gradient and its conjugated double stresses. Instead of introducing an intrinsic material length scale into the constitutive law in an ad hoc fashion, a microstructural granular mechanics approach is applied to derive the higher-order constitutive coefficients such that the internal length scale parameter reflects the natural granularity of the underlying material microstructure. The derivations of the required damage constitutive relationships, the strong form governing equations as well as its weak form for the second gradient model are described. The recently popularized Element-Free Galerkin (EFG) method is then employed to discretize the weak form equilibrium equation for accommodating the resultant higher-order continuity requirements and further handling the mesh sensitivity problem. Numerical examples for shear band simulations show that the proposed second gradient continuum model can produce stable, accurate as well as mesh-size independent solutions without a priori assumption of the shear band path.     

Measurement of the temperature profile during shear band formation in steels deforming at high strain rates

A torsional Kolsky bar (split-Hopkinson bar) was used to deform tubular specimens of AISI 1018 cold rolled steel and AISI 1020 hot rolled steel at a nominal strain rate of 10³s⁻¹. Shear bands were observed to form in both steels, and the temperature of the material in the bands was measured by determining the infrared radiation emitted at the metal surface. For this purpose, a linear array of ten indium-antimonide detectors was used to determine temperature history at ten neighboring points lying across the projected path of the shear band. Results showed that shear bands in these steels are relatively wide, that the maximum temperature rise in the band is about 450°C and that the temperature distribution across the band is consistent with results of stability analyses. The two steels have very different work hardening rates and the strain at which localization is first observed is very different for the two steels : in the cold-rolled steel it occurs at about 15% strain, while in the hot-rolled the strain is near 100%. This result also is consistent with predictions of the analyses.     

Constitutive relations for the shear band evolution in granular matter under large strain

A so-called ＂split-bottom ring shear cell＂ leads to wide shear bands under slow, quasi-static deformation. Unlike normal cylindrical Couette shear cells or rheometers, the bottom plate is split such that the outer part of it can move with the outer wall, while the other part （inner disk） is immobile. From discrete element simulations （DEM）, several continuum fields like the density, velocity, deformation gradient and stress are computed and evaluated with the goal to formulate objective constitutive relations for the powder flow behavior. From a single simulation, by applying time- and （local） space-averaging, a non-linear yield surface is obtained with peculiar stress dependence. The anisotropy is always smaller than the macroscopic friction coefficient. However, the lower bound of anisotropy increases with the strain rate, approaching the maximum according to a stretched exponential with a specific rate that is consistent with a shear path of about one particle diameter.     

动力应变局部化传播及尺寸效应数值模拟

采用 FLAC-3 D模拟了应变软化岩土材料局部化剪切带的发展、扩容对剪切带倾角的影响及试件的宽度效应。随着加载时步的增加 ,彼此孤立的两个应变场逐渐靠近、叠加 ,最终形成了剪切带网络。无论端面约束强或弱 ,剪切带的倾角的数值模拟结果都与罗斯科倾角比较接近。增加试样宽度 ,剪切带的宽度增加 ,剪切带变得不平直 ;宽度越大 ,岩样上端面中点的压应力 -加载时步曲线的峰值强度越大 ,宽度对弹性阶段没有大的影响     

Multiresolution continuum modeling of micro-void assisted dynamic adiabatic shear band propagation

A thermal-mechanical multiresolution continuum theory is applied within a finite element framework to model the initiation and propagation of dynamic shear bands in a steel alloy. The shear instability and subsequent stress collapse, which are responsible for dynamic adiabatic shear band propagation, are captured by including the effects of shear driven microvoid damage in a single constitutive model. The shear band width during propagation is controlled via a combination of thermal conductance and an embedded evolving length scale parameter present in the multiresolution continuum formulation. In particular, as the material reaches a shear instability and begins to soften, the dominant length scale parameter (and hence shear band width) transitions from the alloy grain size to the spacing between micro-voids. Emphasis is placed on modeling stress collapse due to micro-void damage while simultaneously capturing the appropriate scale of inhomogeneous deformation. The goal is to assist in the microscale optimization of alloys which are susceptible to shear band failure.     

Dynamic properties of shear thickening colloidal suspensions

The transient shear rheology (i.e., frequency and strain dependence) is compared to the steady rheology for a model colloidal dispersion through the shear thickening transition. Reversible shear thickening is observed and the transition stress compares well to theoretical predictions. Steady and transient shear thickening are observed to occur at the same value of the average stress. The critical strain for shear thickening is found to depend inversely on the frequency at fixed applied stress for low frequencies (high strains), but is limited to an apparent minimum critical strain at higher frequencies. This minimum critical strain is shown to be an artifact of slip. Lissajous plots illustrate the transition in material properties through the shear thickening transition, and the energy dissipated by a shear thickening suspension is analyzed as a function of strain amplitude.     

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.