Analysis Of Shear Bands Research Articles

A coupled temperature and strain rate dependent yield function for dynamic deformations of bcc metals

A coupled temperature and strain rate microstructure physically based yield function is proposed in this work. It is incorporated along with the Clausius–Duhem inequality and an appropriate free energy definition in a general thermodynamic framework for deriving a three-dimensional kinematical model for thermo-viscoplastic deformations of body centered cubic (bcc) metals. The evolution equations are expressed in terms of the material time derivatives of the elastic strain, accumulated plastic strain (isotropic hardening), and the back stress conjugate tensor (kinematic hardening). The viscoplastic multipliers are obtained using both the Consistency and Perzyna viscoplasticity models. The athermal yield function is employed instead of the static yield function in the case of the Perzyna viscoplasticity model. It is found that the static strain rate value, at which the material shows rate-independent behavior, varies with the material deformation temperature. Computational aspects of the proposed model are addressed through the finite element implementation with an implicit stress integration algorithm. Finite element simulations are performed by implementing the proposed viscoplasticity constitutive models in the commercial finite element program ABAQUS/Explicit [ABAQUS, 2003. User Manual, Version 6.3. Habbitt, Karlsson and Sorensen Inc., Providence, RI] via the user material subroutine coded as VUMAT. Numerical implementation for a simple compression problem meshed with one element is used to validate the proposed model implementation with applications to tantalum, niobium, and vanadium at low and high strain rates and temperatures. The analysis of a tensile shear banding is also investigated to show the effectiveness and the performance of the proposed framework in describing the strain localizations at high velocity impact. Results show mesh independency as a result of the viscoplastic regularization used in the proposed formulation.

On a class of non-coaxial plasticity models for granular soils

The non-coaxiality of the directions of principal stresses and principal plastic strain rates in granular soils under stress rotations has long been observed and recognized in soil tests using both simple shear and hollow cylinder apparatuses. A few constitutive theories have also been proposed in the literature to account for the effect of stress rotations and the subsequent non-coaxial soil behaviour, particularly in the context of shear band analysis. However, the lack of corresponding general numerical methods makes it difficult to investigate the influence of non-coaxial stress–strain behaviour on the results of geotechnical boundary value problems. This paper presents a numerical evaluation of a class of non-coaxial, elastic–plastic models that are developed by combining the conventional plastic potential theory and the double shearing theory. The general non-coaxial constitutive theories are first formulated and then a finite element implementation of the theories is carried out. To evaluate the non-coaxial theories, the problem of simple shear of soils is chosen to investigate the predicted behaviour of soils under simple shear loading conditions where the axes of principal stresses rotate. In particular, the influence of initial stress states and the degree of non-coaxiality are examined. It is found that the numerical results predicted using the non-coaxial model are in general agreement with the experimental observations reported in the literature.

Locally Embedded Strong Discontinuities at finite strains – Numerical Implementation

AbstractIn this paper, a geometrically nonlinear finite element approximation for highly localized deformation in structures undergoing material failure in the form of strain softening, is developed. The basis for its numerical implementation in this class of problems is defined through the elaboration of Strong Discontinuity Approach‐fundamentals. Proposed numerical model uses an Enhanced Assumed Strain Concept for the additive decomposition of the displacement gradient into a conforming and an enhanced part. The discontinuous component of the displacement field which is associated with the failure in the modeled structure is isolated in the enhanced part of the deformation gradient. In contrast to previous works, this part of the deformation mapping is condensed out at the material level, without the application of static condensation technique. The resulting set of constitutive equations is formally identical to that of standard plasticity and therefore, can be solved using the return‐mapping algorithm. No assumptions regarding the interface law connecting the displacement discontinuity with the conjugate traction vector are made. As a result, the proposed numerical solution can be applied to a broad range of different mechanical problems including mode‐I fracture in brittle materials or the analysis of shear bands. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Analysis of shear banding in metallic glasses under bending

An analytical model is proposed for studying the formation of shear bands in metallic glasses over a wide range of size scales when subjected to plane strain bending. The model is based on elasto-plastic beam bending and the evolution of shear offset in shear bands. The spacing of shear bands is assumed to be governed by a scaling law depending on a non-dimensional material parameter which is determined from experiments. The macroscopic inelastic deformation due to bending is thought to be accommodated by slip along shear bands. The model is based on comparison of the energy dissipated along shear bands with the macroscopic dissipation obtained from the elasto-plastic theory of beam bending. The model is able to capture the following experimentally-observed scaling laws: (i) for a given thickness, the shear band spacing varies linearly with curvature; (ii) for a given curvature, the shear band offset varies as the square of the thickness; (iii) the shear band spacing varies linearly with the thickness; and (iv) failure strain (bend ductility) decreases with increasing thickness.

Analysis of adiabatic shear bands in elasto-thermo-viscoplastic materials by modified smoothed-particle hydrodynamics (MSPH) method

We use the modified smoothed-particle hydrodynamics (MSPH) method to analyze shear strain localization in elasto-thermo-viscoplastic materials that exhibit strain- and strain-rate hardening and thermal softening. A homogeneous solution of simple shearing deformations of the body is perturbed and the resulting initial-boundary-value problem analyzed by the MSPH method. It is found that the deformation localizes into a narrow region of intense plastic deformation. In materials exhibiting enhanced thermal softening, an elastic unloading shear wave emanates from this region and propagates outwards. The time when the deformation localizes decreases exponentially with an increase in the thermal softening coefficient. Results have been computed without adding an artificial viscosity and compared with those obtained by the finite element method.

Analysis and Prediction of Shear Banding Under 3D Conditions in Granular Materials

A series of true triaxial tests on tell prismatic specimens of sand at three different void ratios from the basis for analyses, prediction and verification of the influence of shear banding on the three-dimensional failure conditions in granular material. The results of the true triaxial tests are reviewed and discussed in view of the shear banding that is observed in these tests. Theoretical conditions for shear banding, which are based on the behavior of the material near the observed shear banding phenomenon, are reviewed. The detailed procedure for shear banding analysis is explained in general and calculations are performed for a single hardening elasto-plastic constitutive model. The predicted shear banding events are compared with those observed in the experiments. It is shown that the predicted stress conditions for shear banding match those observed in the true triaxial tests, and it is concluded that shear banding plays an important role in the three-dimensional strength of granular materials. Thus, peak failure is caused by shear banding in the hardening regime in the approximate range of b=(σ2-σ3)/(σ1-σ3) form 0.18 to 0.85, while it occurs in the softening regime outside this range of b-values. A smooth, continuous 3D failure surface is therefore not generally obtained for granular materials.

A boundary element technique for incremental, non-linear elasticity: Part I: Formulation

Incremental elastic deformations superimposed upon a given homogeneous strain are analyzed with a boundary element technique. This is based on a recently-developed Green’s function for non-linear incremental elastic deformations. Plane strain perturbations are considered of a broad class of incompressible material behaviours (including hyper-, hypoelastic and Navier–Stokes constitutive equations) within the elliptic range. Numerical treatment of the problem is detailed. A possibility of employing the method in the fully non-linear range is outlined, which yields a boundary element approach where the use of domain integrals is avoided, at least in a conventional sense. The methods for bifurcation and shear band analyses will be reported in Part II.

Discussion of “Analysis of Shear Banding in True Triaxial Tests on Sand” by Poul V. Lade and Qiong Wang

Discussion of “Analysis of Shear Banding in True Triaxial Tests on Sand” by Poul V. Lade and Qiong Wang

Closure to “Analysis of Shear Banding in True Triaxial Tests on Sand” by Poul V. Lade and Qiong Wang

Closure to “Analysis of Shear Banding in True Triaxial Tests on Sand” by Poul V. Lade and Qiong Wang

A two-scale finite element relaxation analysis of shear bands in non-convex inelastic solids: small-strain theory for standard dissipative materials

We propose a fundamentally new approach to the treatment of shear band localizations in strain-softening elastic–plastic solids at small strains based on energy minimization principles associated with micro-structure developments. The point of departure is a general internal variable formulation that determines the inelastic response as a standard dissipative medium. Consistent with this type of inelasticity we consider an incremental variational formulation of the local constitutive response where a quasihyperelastic stress potential is obtained from a local constitutive minimization problem with respect to the internal variables. The existence of this variational formulation allows the definition of the material stability of an inelastic solid based on weak convexity conditions of the incremental stress potential in analogy to treatments in finite elasticity. Furthermore, localization phenomena are interpreted as micro-structure developments on multiple scales associated with non-convex incremental stress potentials in analogy to elastic phase decomposition problems. These micro-structures can be resolved by the relaxation of non-convex energy functionals based on a convexification of the stress potential. The relaxed problem provides a well-posed formulation for a mesh-objective analysis of localizations as close as possible to the non-convex original problem. We develop, based on an approximated rank-one convexification of the incremental stress potential, a computational two-scale procedure for a mesh-objective treatment of localization problems. It constitutes a local minimization problem for a relaxed incremental stress potential with just one scalar variable representing the intensity of the micro-shearing of a rank-one laminate aligned to the shear band. This problem is sufficiently robust with regard to applications to large-scale inhomogeneous deformation processes of elastic–plastic solids. The performance of the proposed energy relaxation method is demonstrated for a representative set of numerical simulations of straight and curved shear bands which illustrate the mesh-independence of the results.

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.

On The Characterization of Plastic Flow in Zr-based Metallic Glass Through Micro-indentation: an Atomic Force Microscopy Analysis

We report an atomic force microscopy analysis of multiple shear bands emerged under multiaxial deformation through micro-indentations of Zr 60 Ni 10 Cu 20 Al 10 metallic glass forming system. Both relative variations of normal and lateral force components as a result of contact between a Si 3 N 4 tip mounted on a cantilever beam and incomplete circular pattern features developed around indentation are recorded at the nano and micro-length scales. Accurate three-dimensional measurements of shear offset as function of position along the shear band and the loading charge in the range of 0.5 to 5 N are determined. Evidence for propagation of discrete displacement increments is shown. Numerical calculations based on the well defined strain localization model including hydrostatic pressure effect on elastic variation of the average free volume are developed to explain the incomplete circular patterns of shear bands observed under constrained deformation.

Analysis of Shear Banding in True Triaxial Tests on Sand

A series of true triaxial tests have been performed on rectangular prismatic specimens of Santa Monica Beach sand at three different relative densities to study the occurrence of failure, mechanisms that create failure, and soil behavior in the vicinity of failure. One mechanism is smooth peak failure, in which the soil continues to behave as a continuum with uniform strains, and smooth peak failure is followed by strain softening. Another mechanism is shear banding, whose occurrence in the plastic hardening regime limits the strength of the soil. Presented here are analyses based on theoretical conditions for localization and subsequent shear banding and on the results of the true triaxial tests. Thus, the strength increases as b [=(σ2 − σ3)/(σ1 − σ3)] increases from 0 to about 0.18, remains almost constant until b reaches 0.85, and then decreases slightly at b = 1.0. Shear banding initiates in the hardening regime for b-values of 0.18–0.85. Thus, peak failure is caused by shear banding in this middle range of b-values, and a smooth, continuous 3D failure surface is therefore not generally obtained for soils. The experimental results agree with the theoretical conditions for the occurrence of shear banding and its consequent effect on the 3D failure stress states for soil.

The analysis of shear banding with a dislocation based gradient plasticity model

A microstructural approach to the incorporation of gradient terms in a single crystal plasticity theory is discussed. A diffusion term that represents cross-slip of dislocations is included in the evolution equations for dislocation densities. A coupling between microscale and macroscale is provided by the dislocation based gradient plasticity model. An algorithmic treatment of the model is given. The effect of the diffusion-like term in the constitutive relation on the resulting formation of localized shear modes is studied. An analysis of a plane strain strip in tension oriented for multiple slip is presented. The effect of the dislocation `diffusivity' on the shear band thickness is analysed.

Analysis of shear bands in slow granular flows using a frictional Cosserat model

ABSTRACTThe slow flow of granular materials is often marked by the existence of narrow shear layers, adjacent to large regions that suffer little or no deformation. This behaviour, in the regime where shear stress is generated primarily by the frictional interactions between grains, has so far eluded theoretical description. In this paper, we present a rigid-plastic frictional Cosserat model that captures thin shear layers by incorporating a microscopic length scale. We treat the granular medium as a Cosserat continuum, which allows the existence of localised couple stresses and, therefore, the possibility of an asymmetric stress tensor. In addition, the local rotation is an independent field variable and is not necessarily equal to the vorticity. The angular momentum balance, which is implicitly satisfied for a classical continuum, must now be solved in conjunction with the linear momentum balances. We extend the critical state model, used in soil plasticity, for a Cosserat continuum and obtain predictions for flow in plane and cylindrical Couette devices. The velocity profile predicted by our model is in qualitative agreement with available experimental data. In addition, our model can predict scaling laws for the shear layer thickness as a function of the Couette gap, which must be verified in future experiments. Most significantly, our model can determine the velocity field in viscometric flows, which classical plasticity-based model cannot.

Localization criteria for non-linear constitutive equations of geomaterials

This paper is devoted to two specific aspects of shear band analysis for geomaterials, which are basically, non-standard materials. Problems arising from incremental non-linearity (multilinear or thoroughly non-linear models) and consequences of discontinuities of the constitutive equations with respect to loading parameters, are especially investigated. Copyright © 2000 John Wiley & Sons, Ltd.

Localization criteria for non‐linear constitutive equations of geomaterials

This paper is devoted to two specific aspects of shear band analysis for geomaterials, which are basically, non-standard materials. Problems arising from incremental non-linearity (multilinear or thoroughly non-linear models) and consequences of discontinuities of the constitutive equations with respect to loading parameters, are especially investigated. Copyright © 2000 John Wiley & Sons, Ltd.

Elasto‐plastic modelling of Red Wildmoor sandstone

On the basis of an extensive experimental program of axisymmetric triaxial compression and extension tests, the mechanical behaviour of a weak sandstone (Red Wildmoor sandstone) is analysed and modelled within the frame of the theory of elastoplasticity. Stress-dependent elasticity and damage are incorporated into the model. Pressure sensitivity of the plastic behaviour is described through a non-linear Mohr–Coulomb model with friction hardening and cohesion softening. Emphasis is given on the calibration procedure. Validation of the model is done through back analysis of the experimental tests. Shear-band analysis is performed on the basis of the extension of the constitutive model to non-coaxial plasticity. Copyright © 1999 John Wiley & Sons, Ltd.

The application of nonlinear gauge mathod to the analysis of local finite deformation in the necking of cylindrical bar

Localized deformation and instability is the focal point of research in mechanics. The most typical problem is the plastic analysis of cylindrical bar necking and shear band under uniaxial tension. Traditional elasto-plastic mechanics of infinitesimal deformation can not solve this problem successfully. In this paper, on the basis of S(strain)-R(rotation) decomposition theorem, the authors obtain the local strain distribution and progressive state of axial symmetric finite deformation of cylindrical bar under uniaxial tension adopting nonlinear gauge approximate method and computer modelling technique.

2D/3D viscoplastic modelling of shear banding

A gradient viscoplasticity model has been used to analyse two- and three-dimensional shear banding. The numerical simulation of a finite width and unique oriention of the shear band is essential. It is demonstrated that the use of viscous regularisation is effective for quasi-static and dynamic problems. The role of the two length scale parameters that are introduced in the model has been investigated. The shear band width is influenced by the two interacting length scales of which the largest dominates the band width. A 3D analysis of shear banding is shown to give significantly different results than the 2D plane strain analysis under similar conditions.