Analysis Of Strain Localization Research Articles

Analysis of strain localization for ductile materials with effect of void growth

In the framework of thermodynamics and damage mechanics, an elastoplastic constitutive relation of ductile materials is developed. Based on this constitutive relation, the properties of strain localization for ductile materials are investigated with the effect of void growth included. The influences of void growth on strain localization are analyzed based on the unified strength criterion. Finally, the effects of yield criteria on the properties of strain localization are discussed.

Strain localization analysis of elasto-viscoplastic soil considering structural degradation

Natural deposits are affected by chemical bonding or the cementation between soil particles during the sedimentation process. Under the effect called `aging', aggregations and linkage between assemblages are formed in the soil structure. It is known that the structure affects the characteristics of the strength, the deformation, and the strain localization. Strain localization is generally considered to be important as a presage of failure. It has recently been recognized that strain localization may occur also during compressive deformation, particularly for porous materials with large confining stress, and the localized compressive deformation may cause large displacement. The aim of the present paper is to study the effect of structural degradation on the localization of both shear and compressive strain for water-saturated clay. Firstly, an elasto-viscoplastic constitutive model considering structural changes with the viscoplastic deformation is proposed in the present paper. Structural collapse is expressed as the shrinking of both the overconsolidation boundary surface and the static yield surface with increasing viscoplastic deformation in the model. Secondly, finite element analyses of undrained and partially drained compression tests under plane strain conditions are conducted to study the effect of structural changes on strain localization. It is seen in the results that the localization of both shear and compressive strain is promoted by the degradation of the soil structure.

Non‐local damage model with evolving internal length

AbstractA modified non‐local damage model with evolving internal length, inspired from micromechanics, is developed. It is shown in particular that the non‐local influence between two points in the damaged material depends on the value of damage at each of these points. The resulting weight function is non‐symmetric and truncated. Finite element results and strain localization analysis on a one‐dimensional problem are presented and compared to those of the original non‐local damage model. It is shown that in the course of damage localization, the incremental strain profiles expand according to the modified non‐local model, instead of shrinking according to the original constitutive relation. Comparisons with experimental data on model materials with controlled porosity are also discussed. Acoustic emission analyses provide results with which the theoretical model is consistent qualitatively. This model also opens the path for durability mechanics analyses, where it has been demonstrated that the internal length in the non‐local model should evolve with environmentally induced damage. Copyright © 2004 John Wiley & Sons, Ltd.

Continuum Damage Approach to Asphalt Concrete Fatigue Modeling

A nonlocal damage model is proposed to predict the behavior of pavement fatigue cracking. This constitutive relation has been implemented in a finite-element code, along with a self-adaptive jump-in-cycle procedure for high cycle fatigue computations. Strain localization analysis shows that during uniaxial fatigue tests, bifurcation due to strain softening occurs much later than in monotonic tests. The incorporation of an internal length into the constitutive model is advocated since the model should encompass loading histories with very different amplitudes of cycles, in which localization may still occur. The influence of the internal length on the fatigue life of bending beams is also investigated. Calibration of the damage model is performed after thermal effects have been evaluated and accounted for in a simplified way, uncoupled to damage. Parameter identification is performed in bending and uniaxial tests. The resulting calibrated constitutive relation is found to yield a good description of several different uniaxial tests.

An analysis of strain localization and wave propagation in plastic models of beams at failure

This paper presents an analysis of localized failures in Timoshenko beams/rods. Standard elastoplastic models are considered first. A spectral analysis of the linearized problem shows the instability of the viscous problem and the ill-posedness of the inviscid rate-independent problem in the presence of strain softening. The consequences of the ill-posedness are illustrated by obtaining the exact solution of wave propagation in a simply supported beam and observing its lack of physical significance: the failure of the beam occurs with no energy dissipation. Finite element solutions of the problem are presented, exhibiting the characteristic pathological mesh-size dependence in these conditions. The lack of a localized dissipative mechanism in these models is identified as the main cause of these inconsistencies, regardless of the length scale present in the problem (namely, the thickness of the beam). To solve these difficulties, we formulate the so-called localized models, incorporating these localized dissipative mechanisms in the form of strong discontinuities in the generalized displacements, leading naturally to the classical notion of plastic hinges. These models consider a localized softening law between the bending moment and the rotation jump. The model problem of wave propagation in a simply supported beam is then reexamined in this context, arriving again at the exact modal solution. This solution recovers its physical significance (the breaking of the beam is modeled with a non-zero energy dissipation), illustrating the well-posedness of the problem and the regularizing effects introduced by the localized models. The comparison of this exact solution with finite element solutions of the problem allows to evaluate the approximation properties of the numerical techniques, showing now the required objective energy dissipation with respect to the mesh size.

Particular aspects of internal length scales in strain localization analysis of multiphase porous materials

Following the work by Zhang et al. [Mech. Cohes.-Frict. Mater. Struct. 4 (1999) 443] we discuss some special features of the internal length scale found in multiphase materials such as saturated and partially saturated porous media, where viscous terms are introduced in the fluid mass balance equations naturally by Darcy's law. Particular attention is focused on the two cases of wave number K=0 and K→∞ of the perturbation waves. The internal length scale properties corresponding to the two cases called “lower and upper critical hardening modulus φ” mentioned in Zhang and Schrefler [Int. J. Numer. Anal. Methods Geomech. 25 (2001) 29], are discussed in detail. It is shown that for the case of the upper value of the critical hardening modulus there will be also a wave number domain for which the material model is dispersive when strain softening behavior occurs for solid skeleton. However, this kind of dispersive waves may not supply enough energy to activate the internal length scale in strain localization analysis. This is true also for the quasi-static case, where it has been found that for both zero and infinite wave numbers the internal length scale associated with permeability disappears. However, it will be pointed out in this paper that the consideration of fluid interaction is necessary for the prediction of the internal length when regularization through a constitutive model is introduced in the numerical model to overcome mesh dependence in a finite element solution. This problem is discussed by considering a gradient dependent model.

Coupled numerical analysis of strain localization induced by shallow tunnels in saturated soils

This paper presents an analysis of shallow tunneling in saturated poro-elastoplastic media by means of an enhanced finite element method with strong discontinuities. To perform a fully coupled analysis of the excavation phase, a commonly adopted procedure for the plane simulation of actual three-dimensional tunneling is extended to the case of pervious cavities in saturated media. The enhanced formulation with strong discontinuities shows to be able to capture the onset of strain localization and its propagation from the tunnel up to the ground surface. These results are contrasted with the unsuccessful simulation of localization obtained by the standard finite element method. The comparison shows that the inability of conventional finite elements in reproducing localization can be a reason for unsatisfying predictions of tunneling induced displacements. Furthermore, the influence of the excavation rate on strain localization, on induced displacements and on tunnel stability is thoroughly investigated with the coupled strong-discontinuity method. The calculated response of the saturated soil is also compared to the results of simulations of tunneling in a dry soil, performed with an uncoupled strong-discontinuity formulation and presented in a previous paper.

Strain localization analysis of elasto-viscoplastic soil considering structural degradation

Natural deposits are affected by chemical bonding or the cementation between soil particles during the sedimentation process. Under the effect called `aging', aggregations and linkage between assemblages are formed in the soil structure. It is known that the structure affects the characteristics of the strength, the deformation, and the strain localization. Strain localization is generally considered to be important as a presage of failure. It has recently been recognized that strain localization may occur also during compressive deformation, particularly for porous materials with large confining stress, and the localized compressive deformation may cause large displacement. The aim of the present paper is to study the effect of structural degradation on the localization of both shear and compressive strain for water-saturated clay. Firstly, an elasto-viscoplastic constitutive model considering structural changes with the viscoplastic deformation is proposed in the present paper. Structural collapse is expressed as the shrinking of both the overconsolidation boundary surface and the static yield surface with increasing viscoplastic deformation in the model. Secondly, finite element analyses of undrained and partially drained compression tests under plane strain conditions are conducted to study the effect of structural changes on strain localization. It is seen in the results that the localization of both shear and compressive strain is promoted by the degradation of the soil structure.

A new algorithm for numerical solution of dynamic elastic–plastic hardening and softening problems

The objective of this paper is to develop a new algorithm for numerical solution of dynamic elastic–plastic strain hardening/softening problems, particularly for the implementation of the gradient dependent model used in solving strain softening problems. The new algorithm for the solution of dynamic elastic–plastic problems is derived based on the parametric variational principle. The gradient dependent model is employed in the numerical model to overcome the mesh-sensitivity difficulty in dynamic strain softening or strain localization analysis. The precise integration method, which has been used for the solution of linear problems, is adopted and improved for the solution of dynamic non-linear equations. The new algorithm is proposed by taking the advantages of the parametric quadratic programming method and the precise integration method. Results of numerical examples demonstrate the validity and the advantages of the proposed algorithm.

An analysis of strain localization in a shear layer under thermally coupled dynamic conditions. Part 1: Continuum thermoplastic models

AbstractThe work presented in this two‐part paper investigates the characteristics of strain localization in a one‐dimensional shear layer with thermomechanical softening behaviour. We consider in this first part the case of a classical continuum model characterized with a stress–strain relation incorporating the inelastic effects through plastic strains. The thermomechanical coupling effects in the infinitesimal shear problem of interest here include the thermal softening of the material with the increase of the temperature, and the heat source generated by the plastic work in the equation of conservation of energy. We first present a spectral analysis of the linearized problem, characterizing its stability and well‐posedness. We consider the general problem including a viscoplastic regularization in this coupled thermomechanical context, without neglecting the elastic effects. This analysis identifies, in particular, the ill‐posedness of the local continuum model under certain conditions, most notably in the inviscid problem with strain softening. The coupled thermal effects are shown then not to regularize the coupled problem in these circumstances. The lack of an internal length scale associated with the strain localization is concluded. The analytical closed‐form solution of the full non‐linear problem is also presented for the identified ill‐posed problems, revealing the lack of physical significance of these models in these cases. The implications of this analysis for the finite element simulations in the form of pathologically mesh‐size dependent solutions is also concluded and illustrated with a number of representative numerical simulations. Copyright © 2003 John Wiley & Sons, Ltd.

An analysis of strain localization in a shear layer under thermally coupled dynamic conditions. Part 2: Localized thermoplastic models

AbstractThe analyses presented in Part 1 of this work have shown the ill‐posedness of the governing equations for a shear layer governed by a coupled thermomechanical continuum model with strain softening. Finite element solutions of the problem have been shown to exhibit a pathological mesh‐size dependence in this case, as a consequence of the aforementioned ill‐posedness. The lack of a localized dissipative mechanism has been traced back to the origin of these difficulties. We consider in this paper the incorporation of such localized dissipative mechanism in the form of a cohesive law along a discontinuity of the displacements, the so‐called strong discontinuity. A coupled thermomechanical model of these discontinuities is formulated involving a continuous temperature field. The discontinuity of the heat flow (i.e. the derivative of the temperature) accommodates the presence of the localized dissipation associated to the thermomechanical cohesive law. In this context, we obtain the exact solution of a problem of wave propagation in the shear layer when the response of the bulk of the layer is elastic. The physical nature of the solution, in contrast with the solutions obtained in Part 1 for continuum models, illustrates the regularizing effect of these localized models, showing in particular the continuous dependence of the solution on the material parameters. Furthermore, we obtain the exact solution of the problem involving a regularized discontinuity in a domain of finite length. This analysis allows to gain a full understanding of the properties of finite element approximations of the problem. Several numerical simulations are presented in this context corroborating the conclusions drawn from the analysis. Copyright © 2003 John Wiley & Sons, Ltd.

Analysis of continental midcrustal strain localization induced by microfracturing and reaction‐softening

Low‐angle extensional shear zones, which often characterize the brittle–ductile transition of the continental crust, are seen here to result from strain localization. The potentially destabilizing deformation mechanism is assumed to be the progressive transformation of fractured coarse feldspar grains into white mica as observed in the East Tenda Shear Zone, Alpine Corsica. The coupling between microfracturing and feldspar‐to‐mica reaction is coeval with strain localization that occurred in that field case at a depth close to 15 km. This reaction is proposed as the main destabilizing factor responsible for the onset of localization, with feldspar having a stationary dislocation creep flow stress larger than mica. To test this hypothesis, a rheological model is constructed based on the field observations for a mixture of three phases—mica, quartz and feldspar—deforming at a common strain rate. The phase concentrations change with time according to the feldspar‐to‐mica reaction, which takes place only if feldspar grains are fractured, a condition detected with the Mohr–Coulomb criterion. The tendency for the strain to localize is assessed by numerical means for the structure composed of an upper crust gliding rigidly over the lower crust, which sustains an overall simple shear. The onset of strain localization is defined by an increase of at least two orders of magnitude in strain rate over part of the lower crust. The upper crust gliding velocity has to be increased by at least a factor of 5 for localization to occur. The time lapse for this velocity change determines the depth of the shear zone (15–17 km). The kinetics of the metamorphic reaction and the final amount of white mica control its width (1–4 km). The time of the shear zone formation is less than half a million years.

４．非線形力学 弾‐熱粘塑性構成式を用いた熱‐水‐土連成有限要素法と飽和粘土の変形解析

It is well known that viscosity of water-saturated clay is affected by temperature variation as well as strain rate. In addition, the conservation of energy during deformation should be considered since the plastic stress power changes the temperature of soil. In this paper, thermohydro-mechanically coupled finite element method using an elasto-thermo-viscoplastic constitutive model for clay is newly developed. And then, a series of strain localization analysis under undrained plane strain conditions and thermal consolidation mainly due to heatgenerated pore water pressure are simulated by thermo-hydro-mechanically coupled finite element method.

A combined parametric quadratic programming and precise integration method based dynamic analysis of elastic-plastic hardening/softening problems

The objective of the paper is to develop a new algorithm for numerical solution of dynamic elastic-plastic strain hardening/softening problems. The gradient dependent model is adopted in the numerical model to overcome the result mesh-sensitivity problem in the dynamic strain softening or strain localization analysis. The equations for the dynamic elastic-plastic problems are derived in terms of the parametric variational principle, which is valid for associated, non-associated and strain softening plastic constitutive models in the finite element analysis. The precise integration method, which has been widely used for discretization in time domain of the linear problems, is introduced for the solution of dynamic nonlinear equations. The new algorithm proposed is based on the combination of the parametric quadratic programming method and the precise integration method and has all the advantages in both of the algorithms. Results of numerical examples demonstrate not only the validity, but also the advantages of the algorithm proposed for the numerical solution of nonlinear dynamic problems.

Analytical and numerical investigation of uniqueness and localization in saturated porous media

AbstractThis paper presents first the applications of uniqueness and strain localization analysis of saturated porous media, where localization of deformation into well defined narrow zones in a saturated porous medium is studied in terms of discontinuous bifurcation theory. A generalized plasticity constitutive model and a Mohr–Coulomb model are used in both the theoretical and numerical analyses of shear band formations. The critical hardening moduli and shear band angle for localization are computed, and quantitative results are given for both constitutive models. Numerical results previously obtained and new ones are confirmed by this analytical and numerical investigation. Copyright © 2002 John Wiley & Sons, Ltd.

Finite element simulation of strain localization with large deformation: capturing strong discontinuity using a Petrov–Galerkin multiscale formulation

The finite element model for strain localization analysis developed in a previous work is generalized to the finite deformation regime. Strain enhancements via jumps in the displacement field are captured and condensed on the material level, leading to a formulation that does not require static condensation to be performed on the element level. A general evolution condition is first formulated laying out conditions for the continued activation of a strong discontinuity. Then, a multiscale finite element model is formulated to describe the post-bifurcation behavior, highlighting the key roles played by the continuous and conforming deformation maps on the characterization of the finite deformation kinematics of a localized element traced by a strong discontinuity. The resulting finite element equation exhibits the features of a Petrov–Galerkin formulation in which the gradient of the weighting function is evaluated over the continuous part of the deformation map, whereas the gradient of the trial function is evaluated over the conforming part of the same map. In the limit of infinitesimal deformations the formulation reduces to the standard Galerkin approximation described in a previous work. Numerical examples are presented demonstrating absolute objectivity with respect to mesh refinement and insensitivity to mesh alignment of finite element solutions.

Numerical analysis of strain localization in metal powder‐forming processes

AbstractIt is well known that strain localization and indeed displacement discontinuity can arise in materials exhibiting plastic behaviour. Indeed such localization is almost certain to occur if strain softening or non‐associated behaviour exists, though it can be triggered even when ideal plasticity is assumed. This study is concerned mainly with the manner in which the numerical discretization process has to be devised so as to capture the localization phenomenon. In this paper, a method is presented for applying the mixed formulation to study the prediction of localization phenomenon in powder‐forming processes. This study is focused on the performance of mixed u−π triangular elements to study their efficiency in indicating localization for various mesh refinements. An adaptive analysis using element elongation is applied in the modelling of strain localization. The numerical results are obtained for a Von Mises yield criterion applied to a multi‐level component, at the final stage of compaction. It is shown that if a correct approximation is used then both the uniform and non‐uniform mesh refinements will converge to the correct answer and clearly indicate the localization phenomenon. It is also observed that an adaptive analysis using element elongation can be effective in the modelling of such phenomena. Copyright © 2001 John Wiley & Sons, Ltd.

Three‐dimensional stochastic finite element method for elasto‐plastic bodies

AbstractA new stochastic finite element method (SFEM) is formulated for three‐dimensional softening elasto‐plastic bodies with random material properties. The method is based on the Karhunen–Loeve and polynomial chaos expansions, and able to efficiently estimate complete probabilistic characteristics of the response, such as moments or PDFs. To reduce the computational complexity in the three‐dimensional setting, two alterations are made with respect to the two‐dimensional SFEM proposed earlier by the authors. First, a variability preserving modification of the Karhunen–Loeve expansion is rigorously derived and applied in the stochastic discretization of random fields representing material properties. Second, an efficient algorithm for parallel processing is developed, with time consumption being the same order as for an ordinary FEM, rendering the proposed SFEM an effective alternative to Monte‐Carlo simulation. The applicability of the proposed method to stochastic analysis of strain localization is examined using Monte‐Carlo simulation. Then, it is applied to a fault formation problem which is a recent concern of earthquake engineering. Ground surface layers are modelled by a softening elasto‐plastic body, and the evolution of probabilistic characteristics of the rupture process is analysed in detail. Some practical observations are made regarding the nature of the fault formation from the stochastic viewpoint. Copyright © 2001 John Wiley & Sons, Ltd.

Analysis of strain localization during tensile tests by digital image correlation

This paper presents an imaging technique developed to study the strain localization phenomena that occur during the tension of thin, flat steel samples. The data are processed using digital speckle image correlation to derive the two in-plane components of the displacement vectors. The authors observe that the calculation of the intercorrelation function reveals a systematic error and propose a numerical method to limit its influence. Plastic incompressibility and thin-sheet assumptions are used to derive the third displacement component and, hence, the various strain and strain rate components. Numerous checks are presented at each step in processing the data to determine the final accuracy of the strain measurements. It is estimated that this accuracy is quite sufficient to track the inception and the development of localization. Examples of possible application are presented for mild steels whose strain localization mechanisms appear to be precocious and gradual.

A finite element model for strain localization analysis of strongly discontinuous fields based on standard Galerkin approximation

This paper presents a finite element model for strain localization analysis of elastoplastic solids subjected to discontinuous displacement fields based on standard Galerkin approximation. Strain enhancements via jumps in the displacement field are captured and condensed on the material level, leading to a formulation that does not require static condensation to be performed on the element level. The mathematical formulation revolves around the dual response of a macroscopic point cut by a shear band, which requires the satisfaction of the yield condition on the band as the same stress point unloads elastically just outside the band. Precise conditions for the appearance of slip lines, including their initiation and evolution, are outlined for a rate-independent, strain-softening Drucker–Prager model, and explicit analytical expressions are used to describe the orientation of the slip line in a plane strain setting. At post-localization the stress-point integration algorithm along the band is exact and amenable to consistent linearization. Numerical examples involving simple shearing of elastoplastic solids with deviatoric plastic flow, as well as plane strain compression of dilatant cohesive/frictional materials, are presented to demonstrate absolute objectivity with respect to mesh refinement and insensitivity to mesh alignment of finite element solutions.