Shear band systems in plane strain extension: analytical solution and comparison with experimental results

Shear banding represents a local failure mechanism of a soil structure as a response to shear loading. In soil structures of different spatial scales systems of regularly spaced shear bands can be observed as a consequence of extensional loading. The phenomenon of single shear bands, defined as thin zones of localized deformation with a discontinuity of the strain field at its boundaries, is well understood. Inside the shear band the material undergoes inelastic strain softening accompanied by shearing and dilation, whereas the material outside the shear band unloads accompanied by elastic contraction in extension tests. Despite numerous experimental and numerical investigations, the physical mechanisms and parameters determining the spacing of parallel shear bands remained unknown. The paper in hand presents an analytical solution for the spacing of the shear bands and a comparison with a large base of experimental data gained from 1g and ng (geotechnical centrifuge) model experiments. The analytical solution is based on the assumption that the elastic energy rate in the unloaded zone between the shear bands tends to a minimum value. The spacing was calculated as the energetically preferred solution for a broad range of cohesive-frictional granular materials. The dependency of the calculated spacing on initial and boundary conditions as well as on material parameters was found to be in good agreement with the experimental results.     

Adiabatic shear banding in a bimetallic body

Summary Thermomechanical deformations of a body made of two different materials and under-going simple shearing deformations are studied with the objectives of finding out when and where adiabatic shear bands will initiate and how they will subsequently grow. Each material is modeled as strain and strain-rate hardening but thermally softening. A shear band is presumed to have formed if the introduction of a temperature perturbation centered around the common interface between the two materials results in an eventual localization of the deformation into a region of width considerably smaller than the width of the initial temperature bump. For a fixed set of material properties the effect of the applied overall strain-rate, and for a fixed applied strain-rate the effect of varying the shear modulus, thermal conductivity, and the coefficient of thermal softening of one material relative to the other have been examined. It is found that a shear band forms in the material that softens more rapidly.With 8 Figures     

剪切带内部应变(率)分析及基于能量准则的失稳判据

应用应变梯度塑性理论对局部化剪切带内部(塑性)剪应变(率)规律进行了理论分析。研究了剪切降模量及岩石材料内部长度等参数对剪切带内部应变(率)的影响。推导了剪应力(率)与剪切带相对错距(速度)的本构关系。研究了剪切降模量和岩石材料内部长度对剪切带稳定性的影响。将岩石试件直剪试验试验机简化为钢块,采用能量准则对岩石试件(剪切带)及钢块系统的稳定性进行了理论研究,提出了系统失稳判据。研究表明:岩石材料的剪切降模量越大,岩石材料的内部长度越小,试验机的剪切刚度越小及试验机的等效高度越大剪切带--钢块系统越容易失稳。     

Effect of material and geometric parameters on deformations near the notch-tip of a dynamically loaded prenotched plate

We analyze plane strain thermomechanical deformations of a prenotched rectangular plate impacted on one side by a prismatic body of rectangular cross-section and moving parallel to the axis of the notch. Both the plate and the projectile are made of the same material. Strain hardening, strain-rate hardening and thermal softening characteristics of the material are modeled by the Johnson–Cook relation. The effect of different material parameters, notch-tip radius, impact speed and the length of the projectile on the maximum tensile principal stress and the initiation and propagation of adiabatic shear bands at the notch-tip is analyzed. It is found that for high impact speeds or enhanced thermal softening, two shear bands, one at −10° to the notch ligament and the other at −128° to it, propagate from the notch tip. Otherwise, only one shear band nearly parallel to the notch-ligament initiates at the notch-tip. The notch-tip distortion for high strength materials is quite different from that for low strength materials. The maximum tensile principal stress occurs at a point on the upper surface of the notch-tip and for every set of values of material parameters and impact speeds studied equals about 2.3 times the yield stress of the material in a quasistatic simple tension or compression test. We assume that the brittle failure occurs when the maximum tensile principal stress equals twice the yield stress of the material in a quasistatic simple tension test and a shear band initiates when the effective plastic strain at a point equals 0.5. The effect of material and geometric parameters on the time of initiation of each failure mode is computed. It is found that for low impact speeds (<30 m/s) a material will fail due to the maximum tensile principal stress exceeding its limiting value, and at high impact speeds due to the initiation of a shear band at the notch-tip. Results are also computed for a C-300 steel with material parameters given by Zhou et al. For an impact speed of 50 m/s, the shear band speed and the maximum effective plastic strain-rate before a material point melts are found to be 350 m/s and 5×10⁷/s respectively. Key words: Failure-mode transition, shear bands, thermoviscoplasticity, numerical simulations. This revised version was published online in July 2006 with corrections to the Cover Date.     

Local and non‐local thermomechanical modeling of elastic‐plastic materials undergoing large strains

This paper deals with the numerical analysis of instabilities for elastic‐plastic materials undergoing large deformations in non‐isothermal conditions. The considered isotropic model is fully thermomechanically coupled and includes temperature‐induced softening, which is another source of strain localization next to geometrical effects. Due to complexity of the model, a symbolic‐numerical tool Ace is used for the preparation of user‐supplied subroutines for the finite element method. The computational verification is performed using two benchmark tests: necking of circular bar in tension and shear banding of elongated rectangular plate in plain strain conditions. The attention is focused on mesh dependence of the numerical results and the regularizing effect of heat conduction. The research reveals that the conductivity influences the shear band width and ductility of the material response; however, for the adiabatic case, the results are discretization sensitive, and another regularization is needed. A new gradient‐enhanced thermomechanical model is developed that introduces an internal length parameter governing the size of the shear band caused by thermal softening. The numerical verification of the non‐local model is performed for the adiabatic case. Subsequently, the simultaneous application of the gradient enhancement and heat conduction in the model is analyzed, which reproduces an evolving shear band. Copyright © 2016 John Wiley & Sons, Ltd.     

A Finite-Deformation Constitutive Model of Bulk Metallic Glass Plasticity

A constitutive model of bulk metallic glass (BMG) plasticity is developed which accounts for finitedeformation kinematics, the kinetics of free volume, strain hardening, thermal softening, rate-dependency and non-Newtonian viscosity. The model has been validated against uniaxial compression test data; and against plate bending experiments. The model captures accurately salient aspects of the material behavior including: the viscosity of Vitreloy 1 as a function of temperature and strain rate; the temperature and strain-rate dependence of the equilibrium free-volume concentration; the uniaxial compression stress-strain curves as a function of strain rate and temperature; and the dependence of shear-band spacing on plate thickness. Calculations suggest that, under adiabatic conditions, strain softening and localization in BMGs is due both to an increase in free volume and to the rise in temperature within the band. The calculations also suggest that the shear band spacing in plate-bending specimens is controlled by the stress relaxation in the vicinity of the shear bands.     

Dynamic shear band development in plane strain compression of a viscoplastic body containing a rigid inclusion

Summary We study the plane strain thermomechanical deformations of a viscoplastic body containing a rigid non-heat-conducting ellipsoidal inclusion at the center. Two different problems, one in which the major axis of the inclusion is parallel to the axis of compression and the other in which it is perpendicular to the loading axis are considered. In each case the deformations are presumed to be symmetric about the two centroidal axes and consequently deformations of a quarter of the block are analyzed. The material of the block is assumed to exhibit strain-rate hardening, but thermal softening. The applied load is such as to cause deformations of the block at an overall strain-rate of 5000 sec^–1. The rigid inclusion simulates the presence of second phase particles such as oxides or carbides in a steel and acts as a nucleus for the shear band.It is found that a shear band initiates near the tip of the inclusion and propagates along a line inclined at 45° to the horizontal axis. At a nominal strain of 0.25, the peak temperature rise near the tip of the vertically aligned inclusion equals 75% of that for the horizontally placed inclusion. The precipitous drop in the effective stress near the inclusion tip is followed somewhat later by a rapid rise in the maximum principal logarithmic strain there.     

Void evolution and coalescence in porous ductile materials in simple shear

The fracture of porous ductile materials subjected to simple shear loading is numerically investigated using three-dimensional unit cells containing voids of various shapes and lengths of the inter-void ligament (void spacing). In shear loading, the porosity reduction is minimal while the void rotates and elongates within the shear band. The strain at coalescence was revealed to be strongly related to the initial void spacing and void shape. It is observed that a transitional spacing ratio for shear coalescence exists with coalescence being unlikely at spacing ratios lower than 0.35. Initially prolate voids are particularly prone to shear coalescence while initially oblate (flat) voids are most resistant to shear failure. The cell geometry becomes sensitive to shear coalescence for increasing void aspect and spacing ratios. In addition, the macroscopic shear stress response becomes independent of the void shape at high spacing ratios while showing a weak dependence on the void shape when the voids are far apart.     

An analysis of shear band development incorporating heat conduction

We incorporate the effects of material temperature sensitivity and heat conduction into an infinite band type analysis of shear localization. Full account is taken of finite geometry changes, but inertial effects are neglected. An energy balance is written between homogeneously deforming bands in a manner that models conditions in our recent finite element study of shear band development from internal inhomogeneities in a solid subject to plane strain compression. The band analysis requires specification of an imperfection amplitude and a length scale over which heat conduction effects are significant. These are chosen to match results of a finite element analysis. The predictions of the simple band analysis and the results of full finite element calculations are then compared for a wide range of material properties for both isotropic and kinematic hardening characerizations of the flow potential surface. The predicted dependence of the onset of locatization on material properties such as strain hardening and strain rate sensitivity is the same for both types of analysis.     

Analysis of stress-induced phase transformations in elastoplastic materials with strain-softening behavior under plane shear

Possible two-phase piecewise-homogeneous deformations in elastoplastic materials with strain-softening behavior under plane shear are analyzed. Discontinuities of stress and deformation gradient across interfaces between phases are considered and continuity of traction and displacement across interfaces and the Maxwell relation is imposed. The governing equations are obtained. The analysis is reduced to finding a minimum value of the loading at which governing equations have a unique, real, physically acceptable solution. It is found that for a plate with constant thickness under plane shear two-phase piecewise-homogeneous deformations are possible, and the Maxwell stress, the stresses and strains in both phases, the jumps of stress and deformation gradient across interfaces and the inclination angle of the localized deformed band can all be determined. As an illustration, a NiTi alloy plate under plane shear is numerically analyzed. The inclination angle of the martensite band is predicted to be 90°, and this predicted value can be applied to explain why no locally deformed spiral martensite band was observed in experiments on thin-walled NiTi alloy tubes under torsion.     

On the role of strain gradients in adiabatic shear banding

Summary The effect of higher order strain gradients on adiabatic shear banding is investigated by considering the simple shearing of a heat conducting thermoviscoplastic material with a gradient-dependent flow stress. The competition between the gradient-dependent plastic flow and heat conduction and their influence on the shear band width and structure are examined. Two internal length scales, i.e., the deformation internal length and the thermal internal length, are incorporated into the linear stability analysis, which shows that the band width size scales either with the square root of the strain gradient coefficient (in the absence of heat conduction) or the thermal conductivity (in the absence of strain gradients), respectively. The numerical computation for the nonlinear problem reveals that the diffusive effect of the strain gradient is much stronger than that of the heat conduction and dictates the constitutive response of the material in the postlocalization regime, and shows that the deformation length scale is much larger than the termal length scale. The band width predicted by the gradient theory agrees reasonably well with the experimental observations found in the literature.     

On adiabatic shear bands in rigid-plastic materials

Summary An analysis is made of a solution obtained by Coleman and Hodgdon in their theory of strain localization in rigid-plastic materials that soften on deformation. It is observed that the solution implies that certain properties of the strain distribution across a shear band should be essentially independent of material parameters and stress level. Among the implications of this type is a general relation which one can use to express the strain at the center of a shear band in terms of the mean strain in the band and the strain outside the band. Theoretical temperature and displacement fields are compared with experimental observations of shear-band formation reported by Costin, Crisman, Hawley, and Duffy for torsional deformation of cold rolled mild steel at ballistic rates.With 3 Figures     

Dynamic shear band development in dipolar thermoviscoplastic materials

We study thermomechanical deformations of a viscoplastic body deformed in plane strain compression at a nominal strain-rate of 5000 sec^-1. We develop a material model in which the second order gradients of the velocity field are also included as kinematic variables and propose constitutive relations for the corresponding higher order stresses. This introduces a material characteristic length l, in addition to the viscous and thermal lengths, into the theory. It is shown that the computed results become mesh independent for l greater than a certain value. Also, the consideration of higher order velocity gradients has a stabilizing effect in the sense that the initiation of shear bands is delayed and their growth is slower as compared to that for nonpolar (l=0) materials.This work was supported by the u.S. Army Research Office grant DAAL03-91-G-0084 and the NSF grant MSS9121279 to the University of Missouri-Rolla. Some of the computations were performed on the Ohio Supercomputer center in Columbus, Ohio     

(001)面双轴应变锗材料的能带调控

本文基于形变势理论构建(001)面双轴应变Ge材料的能带结构模型。计算结果表明(001)面双轴应变可以将Ge的能带从以L能谷为导带底的间接带半导体调控到以Δ4能谷为导带底的间接带半导体或者以Г能谷为导带底的直接带半导体。同时室温下Ge的带隙与应变的关系可用四段函数来表示:当压应变将Ge材料调控为以Г能谷为导带底的间接带半导体后,每增加1%的压应变,禁带宽度将线性减小约78.63meV;当张应变将Ge材料调控为直接带半导体后,张应变每增加1%,禁带宽度将线性减小约177.98meV;应变介于-2.06%和1.77%时,Ge将被调控为以L能谷为导带底的间接带半导体,禁带宽度随着压应变每增加1%而增加11.66meV,随着张应变每增加1%而线性减小约88.29meV。该量化结果可为研究和设计双轴应变Ge材料及其器件提供理论指导和实验依据。     

An adaptive mesh refinement technique for two-dimensional shear band problems

We have developed an adaptive mesh refinement technique that rezones the given domain for a fixed number of quadrilateral elements such that fine elements are generated within the severely deformed region and coarse elements elsewhere. Loosely speaking, the area of an element is inversely proportional to the value of the deformation measure at its centroid. Here we use the temperature rise at a material point to gauge its deformations which is reasonable for the shear band problem since the material within the shear band is deformed intensely and is heated up significantly. It is shown that the proposed mesh refinement technique is independent of the initial starting mesh, and that the use of an adaptively refined mesh gives thinner shear bands, and shaper temperature rise and the growth of the second invariant of the plastic strain-rate within the band as compared to that for a fixed mesh having the same number of nodes. The method works well even when the deformation localizes into more than one narrow region.     

Dynamic shear band development in a thermally softening bimetallic body containing two voids

Summary We study the development of shear bands in a thermally softening viscoplastic prismatic body of square cross-section and containing two symmetrically placed thin layers of a different viscoplastic material and two elliptical voids with their major axes aligned along the vertical centroidal axis of the cross-section. One tip of each elliptical void is abutting the common interface between the layer and the matrix material. Two cases, i.e., when the yield stress of the material of the thin layer in a quasistatic simple compression test equals either five times or one-fifth that of the matrix material are studied. The body is deformed in plane strain compression at an average strain-rate of 5,000 sec^–1, and the deformations are assumed to be symmetrical about the centroidal axes.It is found that in each case shear bands initiate from points on the vertical traction free surfaces where the layer and the matrix materials meet. These bands propagate horizontally into the layer when it is made of a softer material and into the matrix along lines making an angle of ±45° with the vertical when the layer material is harder. In the former case, the band in the layer near the upper matrix/layer interface bifurcates into two bands, one propagating horizontally into the layer and then into other into the matrix material along the direction of the maximum shear stress. The band in the layer near the lower matrix/layer interface propagates horizontally first into the layer and then into the matrix material along the direction of the maximum shear stress. Irrespective of the value of the yield stress for the layer material, a band also initiates from the void tip abutting the layer/matrix interface. This band propagates initially along the layer/matrix interface and then into the matrix material along a line making an angle of approximately 45° with the vertical.     

Determining the stress-strain behaviour at large strains from high strain rate tensile and shear experiments

To characterise the high strain rate mechanical behaviour of metals, split Hopkinson bar experiments are frequently used. These experiments basically yield the force and elongation history of the specimen, reflecting not only the specimen material behaviour but also the specimen structural behaviour. Calculation of the real material behaviour from this global response is not straightforward, certainly for materials such as Ti6Al4V where due to low strain hardening, the specimen deformation is very inhomogeneous. However, for fundamental material research and constitutive material modelling, knowledge of the true effective stress versus plastic strain, strain rate and temperature is essential.In this contribution, a combined experimental-numerical approach for extraction of the strain rate and temperature dependent mechanical behaviour from high strain rate experiments is presented. The method involves the identification of the material model parameters used for the finite element simulations. The technique is applied to determine the stress-strain behaviour of Ti6Al4V using both high strain rate in-plane shear and tensile test results. For the tensile tests, even stress-strain data beyond diffuse necking are retrieved. A comparison is made between the material behaviour extracted from the tensile and the shear experiments. The material behaviour is modelled with the Johnson-Cook constitutive relation. It is found that the simultaneous use of tensile and shear tests to identify the model parameters gives a more generally applicable model. Validation of the material model and the finite element simulations is done by local strain measurements in the shear and tensile test by means of digital image correlation.     

On the gradient-dependent theory of plasticity and shear banding

Summary After a brief review of a recently developed gradient-dependent theory of plasticity various questions related to the yield function and the loading-unloading condition in the presence of higher order strain gradients and the determination of the corresponding phenomenological coefficients are addressed. For rate-independent materials, we construct as before an analytical solution for the strain profile in the postlocalization regime providing the shear band thickness and strain within it but we now compare these results to recently obtained experimental data by assigning appropriate values to the gradient coefficients. We also address some questions recently raised in the literature regarding our nonlinear shear band analysis. For rate-dependent materials, the resulting spatio-temporal differential equation for the strain is solved numerically using the finite difference method. It is shown that the band width does not depend on the grid size, as long as the the grid size is smaller than a certain characteristic length. Various initial imperfections of different amplitudes and sizes are examined, and the possibility of simultaneous development of two shear bands and their interaction is investigated.