Strain Localization Patterns Research Articles

Strain localization in compressed ZrO2(Y2O3) ceramics

Spatiotemporal distributions of local components of the distortion tensor of a nonplastic material—yttria partially stabilized tetragonal zirconia (YTZ) ceramics—have been studied under active compressive straining conditions using double-exposure speckle photography techniques. The strain localization patterns are presented and the features of macroscopic strain inhomogeneity in the elastic state of YTZ ceramics are considered.

Experimental study by pyrometry of Portevin–Le Châtelier plastic instabilities—Type A to type B transition

The dynamic strain aging phenomenon which occurs in some materials under certain temperature and strain rate conditions can cause a localization of the plastic strain in the form of Portevin–Le Châtelier (PLC) bands. According to the strain rate and the temperature, the spatiotemporal pattern of these bands changes and three main types of band are identifiable (types A–C). The increment of plastic strain associated with each band causes an increment in temperature which can be measured by infrared pyrometry. This experimental technique was used in this paper to study the PLC phenomenon in an aluminium–copper alloy at ambient temperature and to visualize the strain localization patterns. Some characteristics of these bands are measured: bandwidth, slope angle, increment of plastic strain in the band, time between two band formations, time of band formation, propagation velocity of the bands, band spacing… The evolution of these characteristics according to the strain enables us to suggest an explanation of the type A–type B transition.

Strain Localization in Solid Cylindrical Clay Specimens Using Digital Image Analysis (DIA) Technique

The strain non-uniformity due to the end restraint for a deforming specimen during triaxial testing of clay specimens was minimized in this study using lubricated end platens. This research presents the evidence of the occurrence of strain localization due to shear banding within the clay specimen by using DIA (Digital Image Analysis) technique. The variation in strain localization patterns of soil specimen is also studied for evaluating the influence of confining stress, loading conditions, stress history, drainage conditions, and soil's microfabric (geometric arrangement of clay platelets) by performing a series of lubricated-end triaxial tests on solid cylindrical specimens of Kaolin clay. This paper presents a comparative study based on the observed orientation of shear band formation, and the intensity of strain localization (by estimating the maximum local strain) within the specimen during its shear deformation process.

Strain localization in geomaterials

Abstract The main purpose of this paper is a broad review of developments in observation and interpretation of localization in geomaterials in the laboratory, with an emphasis on low mean stress situations. Laboratory investigation of strain localization in granular soils and rocks has been pursued extensively and very accurate strain field evolution measurement techniques have been developed, including false relief stereophotogrammetry (FRS) and computed tomography (CT). These permit full characterization of strain localization, from onset to complete shear band formation. This paper reviews studies of sand, clay, sandstone, stiff marl and concrete, and observations of incipient and developed localization in initially ‘homogeneous’ laboratory tests are presented. Development of localization and peak strength, critical stress and strain, shear band orientation and thickness, and complex localization patterns are discussed. Deformation during triaxial compression of sand is shown to develop complex strain localization patterns. Consequently, the critical void ratio concept in granular materials is reconsidered. Void ratio evolution, global and local, monitored by CT, shows a limiting void ratio being rapidly attained in the strain localization zones. In cohesive materials (clays, rocks and concrete), crack development is also commonly observed. Displacement discontinuity measurement techniques are presented and the results for different cohesive geomaterials are discussed.

Strain localization in a continuum as an instability event

The paper presents a novel physical model, based on perturbation theory, to describe localization pattern formation in a solid material as a result of system instabilities. Such kind of approach has been inspired by the theory of population dynamics. In particular, the sinergetic phenomenon of strain localization into a stressed continuum, and its subsequent evolution to cohesive cracking, is obtained through the competition of an external source of energy (e.g., strain energy) and of the internal behavior of the material. The hypothesis of mobile energy entities within material bulk is put forward. These entities, which under low strain conditions are evenly distributed throughout the body, can be considered as strain quanta. The quantization of mechanical quantities is not new in continuum and fracture mechanics, [see, e.g., Novozhilov (1969, Prik Mat Mek 33:212–222)]. With increasing strain, a certain critical point is reached when the homogeneous situation becomes unstable and the strain quanta begin to aggregate into bands, leading to periodic strain localization patterns. The model, which is only theoretical at this stage, can be applied to the particular case of dry snow avalanches. In these cases, snow avalanche triggering is due toinstability (onset of sliding onto a weak plane) and is controlled by external loading (e.g., weight of the slope, load by skiers) and by internal factors (e.g., temperature changes, snow phase transformations etc.).

On dislocation patterning: Multiple slip effects in the rate equation approach

Strain localization and dislocation pattern formation are typical features of plastic deformation in metals and alloys. Glide and climb dislocation motion, along with accompanying production/annihilation processes, lead to the occurrence of instabilities of initially uniform dislocation distributions. These instabilities result to the development of various types of dislocation microstructures (dislocation cells, slip and kink bands, persistent slip bands, labyrinth structures, etc.), depending on the externally applied loading and the intrinsic lattice constraints. The term “dislocation patterning” was introduced over 20 years ago by the third author and a corresponding “gradient dislocation dynamics” framework was suggested to describe such phenomena. In the W–A model proposed at that time by the last two authors, it was shown how coupled nonlinear evolution equations of the reaction-diffusion type for the forest (immobile) and gliding (mobile) dislocation densities can generate dislocation microstructures which correspond to walls perpendicular to the slip direction for Cu-crystals oriented for single slip under cyclic loading conditions. This model is adapted to the multiple slip case here. Weakly nonlinear analysis predicts that dislocation patterns should correspond to domains of walls perpendicular to each slip direction and separated by domain walls in the same orientations. This result is confirmed by numerical analysis and experimental observations. The present model generalizes the original W–A model to the case of multiple slip and considers also explicitly gradient effects by allowing for non-uniform dislocation velocities and internal stress effects.

Modelling fluid transport associated with mineralization and deformation in the Outokumpu Cu–Zn–Co deposit, Finland

Cu–Zn–Co mineralization in the Outokumpu district of eastern Finland shows evidence for structural control, metal enrichment and ore remobilisation during a Paleoproterozoic orogenic event. Our three dimensional numerical models explore deformation and fluid flow patterns within a generic architecture based on the Kylylahti deposit in the Outokumpu district. Patterns of strain localization, dilation and fluid flow associated with the serpentinites and the roof, basal and imbricate thrusts demonstrate the concepts of fluid infiltration and reworking of the Outokumpu deposits during a post-mineralisation deformation event.

Critical state plasticity. Part VI: Meso-scale finite element simulation of strain localization in discrete granular materials

Development of more accurate mathematical models of discrete granular material behavior requires a fundamental understanding of deformation and strain localization phenomena. This paper utilizes a meso-scale finite element modeling approach to obtain an accurate and thorough capture of deformation and strain localization processes in discrete granular materials such as sands. We employ critical state theory and implement an elastoplastic constitutive model for granular materials, a variant of a model called “Nor-Sand”, allowing for non-associative plastic flow and formulating it in the finite deformation regime. Unlike the previous versions of critical state plasticity models presented in a series of “Cam-Clay” papers, the present model contains an additional state parameter ψ that allows for a deviation or detachment of the yield surface from the critical state line. Depending on the sign of this state parameter, the model can reproduce plastic compaction as well as plastic dilation in either loose or dense granular materials. Through numerical examples we demonstrate how a structured spatial density variation affects the predicted strain localization patterns in dense sand specimens.

Modelling deformation of partially melted rock using a poroviscoelastic rheology with dynamic power law viscosity

A poroviscoelastic constitutive model is developed and used to study coupled rock deformation and fluid flow. The model allows the relaxation of both shear and symmetric components of the effective stress. Experimental results are usually interpreted in terms of the power law viscous material. However, in this work the effect of strain damage on viscosity is considered by treating the viscosity as a dynamic time-dependent parameter that varies proportionally to the second invariant of the strain rate. Healing is also taken into account so that the dynamic power law viscosity has a constant asymptotic at a given strain rate. The theoretical model is implemented in a finite element (FE) formulation that couples fluid flow and mechanical equilibrium equations. The FE method is applied to numerically study the triaxial compression of partially melted rocks at elevated PT conditions. It is found that the numerically calculated stress–strain curves demonstrate maxima similar to those observed in laboratory experiments. Also, the computed pattern of melt redistribution and strain localization at the contact between the rock sample and a stiff spacer is qualitatively similar to the experimental observations. The results also indicate that the matrix sensitivity to damage affects the scale of strain localization and melt redistribution.

Non-local damage mechanics with application to concrete

ABSTRACT This paper explains the basic concepts of continuum damage mechanics and its nonlocal formulation. Using one- and two-dimensional examples, it is shown that a stress-strain law with softening postulated within the standard continuum theory leads to physically meaningless results and that the numerical solution suffers by a pathological sensitivity to the finite element discretization. A suitable regularization technique can be based on the non-local formulation, with damage driven by the weighted spatial average of the equivalent strain. Efficiency of the non-local simulation can be increased by mesh-adaptive techniques that adjust the finite element mesh according to the evolving strain localization pattern.

Non-local damage mechanics with application to concrete

ABSTRACT This paper explains the basic concepts of continuum damage mechanics and its nonlocal formulation. Using one- and two-dimensional examples, it is shown that a stress-strain law with softening postulated within the standard continuum theory leads to physically meaningless results and that the numerical solution suffers by a pathological sensitivity to the finite element discretization. A suitable regularization technique can be based on the non-local formulation, with damage driven by the weighted spatial average of the equivalent strain. Efficiency of the non-local simulation can be increased by mesh-adaptive techniques that adjust the finite element mesh according to the evolving strain localization pattern.

On the Portevin–Le Chatelier effect: theoretical modeling and numerical results

A new model is presented for a physically-consistent description of plastic material instabilities referred to as Portevin–Le Chatelier (PLC) effect, namely the oscillatory plastic flow that may be observed in metallic alloys subjected to load-or displacement-controlled plastic deformation in a certain range of strain, strain rate and temperature. The model is conceived to describe the kinetics of Dynamic Strain Ageing (DSA), that is the dynamic interaction between mobile dislocations and diffusing solute atoms which is known to be the primary mechanism inducing the PLC effect. The model is coupled in time and (one-dimensional) space and introduces two intrinsic time scales in the evolution equations and a characteristic length scale through a diffusion-like term with spatial second-order gradient. Approximate analytical solutions are first derived for the boundaries of the PLC range and for the strain localization characteristics defining the kinematics of PLC deformation bands. Numerical results are then obtained through Finite Differences solutions of the space–time coupled equations: a considerable wealth of features is discovered, including the appearance of various PLC band types depending on the applied strain rate. Phenomenological characteristics (i.e. stress–strain curves) are presented together with the corresponding space–time patterns of strain localization. The obtained results are in agreement with the analytical solutions developed here and with the available experimental observations on the PLC effect.

Plastic Flow Localization in Commercial Zirconium Alloys

The behavior of plastic flow curves and patterns of plastic strain localization were studied for tension of samples of Zr — 1% Nb (E110 alloy) and Zr — 1% Nb — 1.3% Sn — 0.4% Fe (E635 alloy) were studied. The relationship of the localization kinetics with the strain hardening law in plastic flow and transition to fracture is established. The dislocation microstructure of the alloys in strain localization and prefracture zones is examined.

Strain localization patterns at a crack tip in generalized single crystal plasticity

Strain localization patterns at a crack tip in generalized single crystal plasticity

Localization of stretching strain in doped carbon γ-Fe single crystals

The evolution of local strain during stretching of high-manganese carbon austenite was studied. The ordered patterns of strain localization proved to be closely related to the stages in the stress-strain curve. The results of this study are compared with analogous data for chromium-nickel nitrogen austenite single crystals. The velocity of self-consistent motion of the sites where plastic strain during stretching of γ-Fe single crystals is nonuniform was determined as a function of the strain hardening coefficient and deformation mechanism.

Instability of gradient-dependent elastoviscoplastic model for clay and strain localization analysis

The instability of the strain gradient-dependent elastoviscoplastic constitutive model for water saturated clay is studied in relation to the strain localization phenomena and pattern formation during deformation. The second-order gradient of volumetric viscoplastic strain is introduced into the constitutive equations to account for the non-local effects associated with the motion of microstructures. Boundary value problems by using finite element are formulated and strain localization analyses are carried out.

Elastic/Crystalline Viscoplastic Finite Element Analysis of Plastic Deformation of FCC Single Crystal Based on Newly Proposed Hardening-Softening Evolution Equation.

Recently, the crystalline plasticity based finite element technology has revealed the great progress in the field of mesomaterial mechanics to study the strength and formability of sheet metal. For this phenomenological mesomaterial modeling, the parameter identification is required by employing the experimental results of single and polycrystal material deformations. In this paper, a hardening-softening evolution equation is newly proposed and embedded in the elastic/crystalline viscoplastic constitutive equation. The uni-axial tension tests of pure aluminum single crystal under the condition of different directions between the crystal orientation [010] and the tensile axis, have provided the stress-strain relationships, slip lines and strain localization patterns. It is demonstrated that numerical results of deformation and strain localization of single crystal sheet tension show good agreements with the experimental observations, which elucidate the anisotropic effect on the deformation clearly.

多孔質非局所弾粘塑性モデルを用いた変形の局所化に及ぼす不均一性と透水性の影響

Deformation behavior of clay specimen modeled as a viscoplastic model with second order strain gradient during shear is numerically analyzed by a soil-water coupled finite element method. In addition, effects of permeability and strain gradient term on growth rate of fluctuation were obtained by the linear instability analysis. It isfound that the strain localization pattern and stress-strain curve greatly depend on the initial inhomogeneity and the permeability.

Cosserat Modeling of Size Effects in Crystalline Solids

AbstractThe mechanics of generalized continua provides an efficient way of introducing intrinsic length scales into continuum models of materials. A Cosserat framework is presented here to descrine the mechanical behavior of crystalline solids. The first application deals with the problem of the stress field at a crak tip in Cosserat single crystals. It is shown that the strain localization patterns developping at the crack tip differ from the classical picture : the Cosserat continuum acts as a bifurcation mode selector, whereby kink bands arising in the classical framework disappear in generalized single crystal plasticity. The problem of a Cosserat elastic inclusion embedded in an infinite matrix is then considered to show that the stress state inside the inclusion depends on its absolute size lc. Two saturation regimes are observed : when the size R of the inclusion is much larger than a characteristic size of the medium, the classical Eshelby solution is recovered. When R is much small than the inclusion, a much higher stress is reached (for an inclusion stiffer than the matrix) that does not depend on the size any more. There is a transition regime for which the stress state is not homogeneous inside the inclusion. Similar regimes are obtained in the study of grain size effects in polycrystalline aggregates of Cosserat grains.

Instability of gradient dependent elasto-viscoplasticity for clay

A gradient-dependent viscoplastic constitutive model for water saturated clay is proposed to describe the strain localization phenomena and pattern formation during deformation. Second- and fourth-order gradients of volumetric viscoplastic strain are introduced into the constitutive equations to account for the non-local effects due to the motion of microstructures. A linear perturbation analysis is applied to this model. The instability of the government equations (i.e. the constitutive equations and the equations of motion for the clay skeleton and pore water) is discussed for both the one-dimensional and the two-dimensional situations. In addition, issues concerned with the formulation of boundary value problems by finite element analysis in relation to the formulation and the boundary conditions are presented.