A finite element analysis of strain localization for soft rock using a constitutive equation with strain softening

Summary It is well known that deformation and/or strain of geological materials localize when they come close to the failure state. In the present: study, we try to analyze the deformation problem using a constitutive relation with strain hardening and strain softening. The constitutive model of a soft rock and overconsolidated clay using an elasto-plastic constitutive theory with memory was originally developed by Adachi and Oka. This type of formulation is shown to be easily applied to analyze the material behavior of strain softening because there is a similarity to that in viscoplasticity. Using the proposed model, the loss of uniqueness of the solution to the initial value problem can be avoided and a special or complicated numerical technique, e.g., an arc length method, does not need to be used. When we use constitutive equations with strain softening in a finite element analysis, there is a problem of strong mesh size dependency of numerical results. To remedy the mesh size dependency, we generalize the Adachi-Oka model based on the concept of non-localization by Bazant. We apply the proposed constitutive model to the behavior of a sedimentary soft rock in the drained triaxial compression test. It is found that mesh size dependency becomes smaller using the non-localization of the constitutive model.

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Eine Finite-Element-Analyse der lokalisierten Verformung von weichem Gestein bei einem Stoffgesetz mit Entfestigung

Übersicht Bekanntlich findet bei geologischen Stoffen in der Nähe des Versagenszustandes eine Lokalisierung der Verformung statt. Die Analyse dieses Problems wird in diesem Beitrag auf der Grundlage eines Stoffansatzes mit Ver- und Entfestigung unternommen. Entwickelt wurde das zugrunde gelegte elastischplastische Stoffmodell mit Gedächtnis von Adachi und Oka für weiches Gestein und übermäßig verdichteten Ton. Wegen der Ähnlichkeit zur Viskoplastizität läßt sich diese Formulierung des Stoffgesetzes leicht auf die Analyse des Verhaltens von entfestigendem Material anwenden, da der Eindeutigkeitsverlust der Lösung des Anfangswertproblems vermieden wird und besondere Rechenverfahren wie etwa die Bogenlängenmethode nicht benötigt werden. Bei der Benutzung von Stoffgesetzen in einer Finite-Element-Rechnung hängen die Ergebnisse stark von der Netzeinteilung ab. Um dies abzustellen, wird das Modell von Adachi und Oka auf der Grundlage von Bazant's Konzept der Nichtlokalisierung verallgemeinert. Anwendungsbeispiel ist das Verhalten von weichem Gestein im drainierten Triaxial-Test. Es zeigt sich, daß mit dem Konzept der Nichtlokalisierung im Stoffmodell der Einfluß der Netzeinteilung geringer wird.

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Presented at the workshop on Numerical Methods for Localization and Bifurcation of Granular bodies, held at the Technical University of Gdask (Poland), September 25–30, 1989.     

A viscoplastic model combined smedamage and smeared crack for softening of concrete

A viscoplastic model accounting for developing damage in concrete is proposed by assuming the rate of damage to be dependent on viscous strain and stress rates. The damage is measured by a scalar parameter affecting both the yield stress and the material viscosity. For a post-critical range of deformation, the localized mode occurs for which additional constitutive equations are specified. The model is applied to simulate uniaxial strain rate controlled and creep response for the concrete.     

Texture and strain localization prediction using a N-site polycrystal model

Most polycrystal models of plastic deformation rely on the assumption that strain and stress are uniform within the domain of each grain. Comparison between measured and predicted textures suggests that this assumption is realistic for most single-phase aggregates and crystal symmetries. In this paper, we implement a self-consistent N-site model that allows one to account for strain localization and local misorientation near grain boundaries. We apply this model to face centered cubic (fcc) and hexagonal close packed (hcp) aggregates, and analyze the similarities and differences with a one-site model that assumes uniform stress and strain-rate within a grain. We find that the assumption of uniformity is justified in first order. We discuss the implications of the N-site model for the simulation of systems with hard inclusions, orientation correlations, and recrystallization mechanisms.     

A strain space nonlinear kinematic hardening/softening plasticity model

A strain space plasticity theory based on the nonlinear kinematic hardening and softening rule is developed in order to accommodate work-hardening, work-softening, and elastic-perfectly plastic materials with one set of constitutive equations, and to facilitate strain controlled calculations. A generalized hardening/softening parameter is proposed, and the potential of linking the parameter to micro-mechanical material changes is discussed. The theory is used to investigate work-softening materials numerically and highlights a need for additional experimental results in this area.     

Inelastic analysis of plastic compressible materials using the viscoplastic model

In this study the theory of viscoplasticity is employed to provide a unified approach to the problems of plasticity and creep. For plastic compressible materials, the viscoplasticity equation is formulated and the effective range of the equation is discussed. The Drucker-Prager potential is used to describe the plastic compressible yielding and the strength-differential effect. The influence of the strength-differential effect is investigated for a pressurized cylinder.     

A viscoplastic strain gradient analysis of materials with voids or inclusions

A finite strain viscoplastic nonlocal plasticity model is formulated and implemented numerically within a finite element framework. The model is a viscoplastic generalisation of the finite strain generalisation by Niordson and Redanz (2004) [Journal of the Mechanics and Physics of Solids 52, 2431–2454] of the strain gradient plasticity theory proposed by Fleck and Hutchinson (2001) [Journal of the Mechanics and Physics of Solids 49, 2245–2271]. The formulation is based on a viscoplastic potential that enables the formulation of the model so that it reduces to the strain gradient plasticity theory in the absence of viscous effects. The numerical implementation uses increments of the effective plastic strain rate as degrees of freedom in addition to increments of displacement. To illustrate predictions of the model, results are presented for materials containing either voids or rigid inclusions. It is shown how the model predicts increased overall yield strength, as compared to conventional predictions, when voids or inclusions are in the micron range. Furthermore, it is illustrated how the higher order boundary conditions at the interface between inclusions and matrix material are important to the overall yield strength as well as the material hardening.     

A microstructure-based viscoplastic model for asphalt concrete

A microstructure-based viscoplastic continuum model is developed for the permanent deformation of asphalt concrete (AC). The model accounts for several phenomena that influence the permanent deformation of AC at high temperatures. These phenomena include strain rate dependency, confining pressure dependency, dilation, aggregate friction, anisotropy, and damage.The material anisotropy was included in the model by replacing the stress invariants in the yield function with invariants of both the stress and a microstructure tensor, which describes the aggregate orientation distribution. The components of the microstructure tensor were determined using image analysis techniques (IAT) conducted on digital images taken from two-dimensional cut sections of AC. Furthermore, damage was included in the model based on the effective stress theory to account for crack and air void growth that significantly reduces the load-carrying capacity of the material.Experimental data from triaxial compressive strength tests conducted at five strain rates and three confining pressures were used to develop a methodology to determine the material parameters that characterize AC permanent deformation. The model predictions were in a good agreement with the experimental measurements.     

脆性固体损伤和局部软化的有限元分析

本文研究混凝土、岩石一类工程中常用的应变软化材料的有限元分析方法。在作者以往有关粘塑性损伤本构模型的工作基础上，给出了一组便于有限元计算的本构方程表达式。包括损伤弹性矩阵和局部损伤软化矩阵，分别运用于计算硬化和软化阶段的有限元刚度矩阵；对所提出的本构方程的实验验证计算和一些算例的有限元数值分析，表明文中给出的本构方程是可行的，相应的有限元算法能较好地对损伤固体的局部软化效应进行数值分析，并可成功地追踪应力应变响应的软化曲线     

A viscoplastic constitutive model incorporating dynamic strain aging effect during cyclic deformation conditions

In this paper, a viscoplastic constitutive model previously proposed by the authors was extended to apply to the cyclic deformation analysis of the modified 9Cr-1Mo steel. A series of cyclic deformation tests were conducted on modified 9Cr-1Mo steel at various temperatures, including those under anisothermal conditions. Furthermore, cyclic evolution of state variables used in the authors' constitutive model was experimentally measured. Based on the test results, cyclic softening behavior depending on the temperature and its history was introduced into the constitutive model. The extended model was applied to simulations of inelastic deformation behavior under monotonic tension, stress relaxation, creep, isothermal cyclic deformations including stress relaxation and anisothermal cyclic deformations. It was found that the present constitutive model has a capability of describing the inelastic deformation behavior of modified 9Cr-1Mo steel adequately at various loading conditions.     

Limit analysis of viscoplastic thick-walled cylinder and spherical shell under internal pressure using a strain gradient plasticity theory

Plastic limit load of viscoplastic thick-walled cylinder and spherical shell subjected to internal pressure is investigated analytically using a strain gradient plasticity theory. As a result, the current solutions can capture the size effect at the micron scale. Numerical results show that the smaller the inner radius of the cylinder or spherical shell, the more significant the scale effects. Results also show that the size effect is more evident with increasing strain or strain-rate sensitivity index. The classical plastic-based solutions of the same problems are shown to be a special case of the present solution.     

Post-critical discontinuous localization analysis of small-strain softening elastoplastic solids

Summary The paper presents some aspects of the formulation and numerical implementation of localization phenomena in elastoplastic solids at small strains. At first we point out the theoretical foundation and algorithmic implementation of a localpre-critical localization analysis for the detection of critical zones which may cause a loss of global structural stability, e.g. shear-bands. The second part proposes a particular approach to thepost-critical localization analysis in order to trace post-critical localized equilibrium branches of solids with a global displacement-discontinuity surface. Within this context we investigate two general constitutive approaches to the modelling of the displacement discontinuity and develop a finite-element formulation for its numerical implementation.

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Postkritische diskontinuierliche Lokalisierungsberechnung von erweichenden elastoplastischen Körpern bei kleinen Verzerrungen

Übersicht Die Arbeit stellt Aspekte der Formulierung und numerischen Implementation von Lokalisierungsphänomenen in elastoplastischen Festkörpern im Rahmen kleiner Verzerrungen dar. Zunächst diskutieren wir die theoretischen Grundlagen und die algorithmische Implementation einer lokalen begleitendenprä-kritischen Lokalisierungsanalysis zur Aufdeckung von kritischen Zonen wie Scherbändern, die den Verlust der globalen Strukturstabilität bewirken können. Der zeiite Teil schlägt einen besonderen Zugang zu einerpost-kritischen Lokalisierungsanalysis vor zur Verfolgung postkritischer lokalisierter Gleichgewichtspfade von Strukturen mit einer globalen Verschiebungsdiskontinuität. In diesem Zusammenhang untersuchen wir zwei allgemeine konstitutive Zugänge zur Modellierung der Verschiebungsdiskontinuität und entwickeln eine Finite-Elemente-Formulierung zu ihrer numerischen Implementation.

     

A homogenization-based constitutive model for isotropic viscoplastic porous media

An approximate model based on the “second-order” nonlinear homogenization method is proposed to estimate the effective behavior of isotropic, viscoplastic, porous materials. The model is constructed in such a way that it reproduces exactly the behavior of a “composite-sphere assemblage” in the limit of hydrostatic loadings, and therefore coincides with the hydrostatic limit of Gurson’s criterion in the special case of ideal plasticity. As a consequence, the new model improves on earlier homogenization estimates, which have been found to be quite accurate for low triaxialities but overly stiff for sufficiently high triaxialities and nonlinearities. Additionally, the estimates delivered by the model exhibit a dependence on the third invariant of the macroscopic stress tensor, which has a nontrivial effect on the effective response of the material at moderate triaxialities. The proposed model is compared with exact results obtained for a special class of porous materials with sequentially laminated microstructures. The agreement is found to be quite good for the entire range of stress triaxialities, and all values of the porosity and nonlinearity considered.     

基于非局部塑性模型的应变局部化理论分析及数值模拟

通过求解一个第二类Fredholm方程,得到了基于非局部塑性软化模型的应变局部化问题理论解,结果表明,只有在当采用过非局部修正形式的非局部塑性软化模型才能得到应变局部化解,且得到的塑性应变分布和荷载响应依赖于所引入的特征长度及过非局部权参数。通过一维应变局部化有限元数值解,验证了非局部理论的引入能克服计算结果的网格敏感...     

A viscoplastic model based on no-yield-surface concept

The elastic/viscoplastic constitutive equation which describes deformation law of metal materials was suggested based on no-yield-surface concept and thermal activation theory of dislocation. The equation which takes account of effects of strain-rate, strain history, strain-rate history, hardening and temperature has stronger physical basis. Comparison of the theoretical prediction with experimental results of mechanical behaviours of Ti under conditions of uniaxial stress and room temperature shows good consistency.The Projects Supported by National Natural Science Foundation of China     

Finite element simulations of martensitic phase transitions and microstructures based on a strain softening model

A new approach for modeling multivariant martensitic phase transitions (PT) and martensitic microstructure (MM) in elastic materials is proposed. It is based on a thermomechanical model for PT that includes strain softening and the corresponding strain localization during PT. Mesh sensitivity in numerical simulations is avoided by using rate-dependent constitutive equations in the model. Due to strain softening, a microstructure comprised of pure martensitic and austenitic domains separated by narrow transition zones is obtained as the solution of the corresponding boundary value problem. In contrast to Landau-Ginzburg models, which are limited in practice to nanoscale specimens, this new phase field model is valid for scales greater than 100 nm and without upper bound. A finite element algorithm for the solution of elastic problems with multivariant martensitic PT is developed and implemented into the software ABAQUS. Simulated microstructures in elastic single crystals and polycrystals under uniaxial loading are in qualitative agreement with those observed experimentally.     

A viscoplastic constitutive model with effect of aging

It is recognized experimentally that differences in plastic flow stress due to the change in strain rate of SUS304 stainless steel are found to decrease after cyclic preloadings, but minimal change is observed in relaxation properties. Therefore, viscosity function based on tensile stress–strain properties differs from that obtained from relaxation behavior. For such case, the existing visco-plastic constitutive concept, such as the so-called overstress model where only viscosity is taken into consideration, has a poor capability in predicting the time-dependent mechanical properties systematically. A new viscoplastic constitutive concept is presented to analyze the phenomenon. In the constitutive concept, the dynamic strain aging even at room temperature, as well as viscosity, are introduced as the dominant factors of the time-dependent plastic deformation. An experimental technique is proposed and some experimental results are presented to estimate the effects of aging and viscosity separately on the time-dependency of a SCM435 low alloyed steel under tensile loading. The proposed constitutive model with aging is verified for the systematical predictions of both plastic flow properties and relaxation behavior of the SCM435 low alloyed steel.     

On a viscoplastic Shanley-like model under constant load

Motivated by applications devoted to study the behavior of steel and aluminum alloys columns, inelastic Shanley-like models have been extensively studied in the literature, mainly to investigate buckling and post buckling problems (see Sewell, 1971; Hutchinson, 1974 for a complete review) .On the other hand, recent papers discussing geotechnical problems point out that those models may be useful for the study of the essential features of the equilibrium of towers. In this case, the structures proper weight (which is a conservative load with constant magnitude) , and the verticality imperfection, appear to be responsible for the leaning evolution, as well as the time variation of the mechanical property of the soil.Throughout this paper, a T shaped rigid rod on two no-tension viscoplastic springs under constant load with initial imperfection is considered. Under fairly general assumptions, a viscoplastic constitutive law is derived as a particular case of the theory developed in (Gurtin et al., 1980) , studying its behavior under loading processes. By virtue of a time rescaling procedure, extreme retardation leads to determine a yielding parameter, which allows to distinguish between viscoelastic and viscoplastic ranges.For all the states attained by the rod, explicit expressions for the two displacement parameters characterizing its evolution are given. Noting that failure may occur if the reaction of one spring goes to zero, sufficient conditions under which no bifurcation and no failure occur are given for all the phases, leading so to determine the minimum upper bound for the load parameter. This new result turns out to depend only on the relaxation surface parameters at equilibrium, irrespective of the behavior under non-zero finite deformation velocities.