Associative Flow Rule Research Articles

Performance Dependent Model for normal and high strength concretes

A new constitutive formulation, the so-called Performance Dependent Model valid for normal and high strength concretes is presented. The distinctive aspect of the proposed model is the consideration of relevant properties of concrete mix components in the evaluation of the involved material performance or quality at the macroscopic stand point. In this way, the composite features of concrete are appropriately taken into account. The model maximum strength surface is defined by means of the Performance Dependent Failure Criterion proposed by the authors in previous works. Concrete behaviors in pre and post peak regimes are modeled with a non uniform hardening law and an isotropic softening rule, respectively. To realistically reproduce the concrete ductility in pre and post peak regimes under different load scenarios, the hardening and softening laws are defined in terms of the acting confining pressure. Concrete dilatancy behavior is approached by means of a volumetric non associative flow rule. The softening law is embedded in fracture energy concepts for mode I and II types of failure. The model considers two main input material parameters: the uniaxial compressive strength and the performance parameter, a quality index defined in the context of the Performance Dependent Failure Criterion. The proposed constitutive model is able to capture the substantial differences in the failure behavior of normal and high strength concretes as well as of concretes with the same compressive strength but different mix components. The predictive capability of the model is demonstrated in the numerical analyses included in this paper where the numerical predictions are compared with experimental results related to concrete specimens of different qualities and subjected to stress histories under both compressive and tensile regimes.

Slope Stability Analysis Using Unified Strength Theory

Numerical computations using finite difference method and unified strength theory are reported to analyze the slope stability problem. The Factor of safety of plane strain and axisymmetric slopes was calculated by strength reducing method, and the influences of intermediate principal stress on slope stability problem was analyzed. The associative and non-associative flow rule was taking into account in plane strain slope problem analysis. The intermediate principal stress has equivalent influences on slope stability problem under associative and non-associative flow rule. The Factor of safety of plane strain slope is lower than the axisymmetric situation. The influence of intermediate principal stress on slope stability under plane strain condition is heavier than axisymmetry.

Deriving Plastic Constitutive Equations of the Material in Complex Stress State by Means of Simple Test Results

By conducting simple tension and torsion tests to material, constitutive equations of one dimension are obtained. Plastic theories of continuum mechanics are used for analyzing deformation behavior of the material after yielding. Here, material is presumed to have isotropic hardening characteristic. By using Mises loading function and the associative flow rule, the derivations are made to extend the constitutive equations of one dimension in the simple tension and torsion tests to that of multi-dimension and obtain the plastic constitutive equations of the material in complex stress state , respectively.

Cyclic plasticity modeling and finite element analyzes of a circumferentially notched round bar under combined axial and torsion loadings

A cyclic plasticity model suitable for fatigue damage modeling of metallic structures is presented and its finite element (FE) implementation is described within the small strain plasticity framework. The model uses the von Mises yield surface in stress space whose evaluation follows an Armstrong–Frederick type of nonlinear kinematic hardening rule, and the normality hypothesis in conjunction with the associative flow rule is assumed. An incremental implicit-iterative algorithm was employed for the numerical solution of resulting stress–strain equations, and the continuum tangent obtained from plasticity model was used in FE implementation. The developed FE computational model is applied in the cyclic deformation analysis of a circumferentially notched specimen in combined axial force-torsion loading tests. The computed notch root deformations were compared with measured notch root strain histories. An assessment of model predictions showed that non-proportional loading tests have been simulated with a good accuracy. The computed strain loops were in accord with experimental data and matched qualitatively with measured shear – axial strain histories irrespective of loading path of the test. In proportional balanced torsion-axial loading, the nonlinear shear strain – axial strain loops were also simulated properly. The errors in notch root strains were dependent on the loading path shape, and compared to axial strains, the shear strain errors were relatively greater. The computer solution times were also acceptable.

Analytical and numerical analyses of the load‐bearing capacity of retaining walls laterally supported at both ends

AbstractThis paper investigates the load‐bearing capacity of a perfectly smooth retaining wall laterally supported at both ends assuming that the wall fails by the development of three plastic hinges. The study considers the case of a cohesionless elastic–perfectly plastic backfill with a Mohr–Coulomb yield criterion and an associative flow rule in drained conditions. A kinematically admissible soil–structure failure mechanism is proposed and compared with the conventional solutions and with results from a numerical finite element modelling. The study shows that the proposed solution and the numerical solution are in good agreement. These solutions are found to be much more favourable for the wall than the conventional solutions. Copyright © 2010 John Wiley & Sons, Ltd.

Numerical simulation of ram extrusion in short-fiber-reinforced fresh cementitious composites

A series of ram extrusion tests was carried out on a short-fiber-reinforced, semisolid, fresh cementitious composite. An elastoviscoplastic constitutive model is proposed for the extrudable fresh cementitious composite. It features the associative flow rule, a nonlinear strain rate-hardening law, and the von Mises yield criterion. The model is then implemented in ANSYS/LS-DYNA explicit finite element code. Various ram extrusion processes of the fresh cementitious composite were simulated. It has been found that the extrusion load versus imposed displacement predictions agree well with the experimental results. The fresh paste flow, through the die entry and the die-land, is then interpreted in light of the evolution of the deformation and distribution of state variables, mainly based on numerical results and the ram extrusion mechanism. The effects of extrusion ratio and extrusion velocity on extrusion load are also investigated, based on the mechanical properties of the fresh cementitious composite. The study indicates that the numerical procedure established, together with the constitutive model proposed, is applicable for describing ram extrusion of short-fiber-reinforced fresh cementitious composites, which might provide a numerical rheometric tool from which ram extrusion of elastoviscoplastic paste-like materials can be examined and quantified.

Numerical study of the bearing capacity for two interfering strip footings on sands

Abstract Current studies of bearing capacity for shallow foundations tend to rely on the hypothesis of an isolated footing. In practice a footing is never isolated; it is mostly in interaction with other footings. This paper focuses on a numerical study using the finite-difference code Fast Lagrangian Analysis of Continua (FLAC), to evaluate the bearing capacity for two interfering strip footings, subjected to centered vertical loads with smooth and rough interfaces. The soil is modeled by an elasto-plastic model with a Mohr–Coulomb yield criterion and associative flow rule. The interference effect is estimated by efficiency factors, defined as the ratio of the bearing capacity for a single footing in the presence of the other footing to that of the single isolated footing. The efficiency factors have been computed individually to estimate the effects of cohesion, surcharge, and soil weight using Terzaghi’s equation, both in a frictional soil with surcharge pressures and in a cohesive-frictional soil with surcharge pressures. The results have been compared with those available in the literature.

A destructuration theory and its application to SANICLAY model

Many natural clays have an undisturbed shear strength in excess of the remoulded strength. Destructuration modeling provides a means to account for such sensitivity in a constitutive model. This paper extends the SANICLAY model to include destructuration. Two distinct types of destructuration are considered: isotropic and frictional. The former is a concept already presented in relation to other models and in essence constitutes a mechanism of isotropic softening of the yield surface with destructuration. The latter refers to the reduction of the critical stress ratio reflecting the effect of destructuration on the friction angle, and is believed to be a novel proposition. Both the types depend on a measure of destructuration rate expressed in terms of combined plastic volumetric and deviatoric strain rates. The SANICLAY model itself is generalized from its previous form by additional dependence of the yield surface on the third isotropic stress invariant. Such a generalization allows to obtain as particular cases simplified model versions of lower complexity including one with a single surface and associative flow rule, by simply setting accordingly parameters of the generalized version. A detailed calibration procedure of the relatively few model constants is presented, and the performance of three versions of the model, in descending order of complexity, is validated by comparison of simulations to various data for oedometric consolidation followed by triaxial undrained compression and extension tests on two structured clays. Copyright © 2009 John Wiley & Sons, Ltd.

Convexity of Yield Surface with Directional Distortional Hardening Rules

The present paper examines the convexity of the yield surface in the directional distortional hardening models by Feigenbaum and Dafalias. In these models anisotropy develops through kinematic and directional distortional hardening, supplemented by the classical isotropic hardening, and the associative flow rule is used. However, the issue of convexity, which naturally arises due to the distortion of the yield surface, was not fully addressed. The present paper derives the necessary and sufficient conditions to ensure convexity of the yield surface for the simpler Feigenbaum and Dafalias models, but it is not as straightforward to derive corresponding conditions for convexity of the Feigenbaum and Dafalias model version which contains an evolving fourth-order tensor. In this case convexity will be addressed first in general and then at the limit state for which simple restrictions on the material constants to ensure convexity are derived. Numerical examples will show that some of the yield surfaces simulated in the original Feigenbaum and Dafalias publication will not stay convex if loaded beyond what was done in these publications. Therefore the material constants for these cases are recalibrated based on the derived relations for satisfaction of the convexity requirement, and the fitting of the yield surfaces is repeated with the new set of constants and compared with the previous case.

3-dimensional slope stability analyses using non-associative stress-strain relationships

The research work presented in this paper refers to a new slope stability analysis method used for landslide risk evaluations. It is an extension of the 3-dimensional upper-bound slope stability analysis method proposed by Chen et al. in 2001, which employs the Mohr-Coulomb’s associative flow rule. It has been found that in a 3-dimensional area, a prism may not be able to move at friction angles to all its surrounding interfaces, as required by this associative rule, and convergence problems may occasionally arise. The new method establishes two velocity fields: (i) The plastic one that represents a non-associative and the best representative dilation behavior, and (ii) the virtual one that permits the solution for factor of safety in the work and energy balance equation. The new method can then allow any input value of dilation angle and thus solve the convergence problem. A practical application to a concrete dam foundation is illustrated.

Anisotropic hardening and non-associated flow in proportional loading of sheet metals

Conventional isotropic hardening models constrain the shape of the yield function to remain fixed throughout plastic deformation. However, experiments show that hardening is only approximately isotropic under conditions of proportional loading, giving rise to systematic errors in calculation of stresses based on models that impose the constraint. Five different material data for aluminum and stainless steel alloys are used to calibrate and evaluate five material models, ranging in complexity from a von Mises’ model based on isotropic hardening to a non- associated flow rule (AFR) model based on anisotropic hardening. A new model is described in which four stress–strain functions are explicitly integrated into the yield criterion in closed form definition of the yield condition. The model is based on a non-AFR so that this integration does not affect the accuracy of the plastic strain components defined by the gradient of a separate plastic potential function. The model not only enables the elimination of systematic errors for loading along the four loading conditions, but also leads to a significant reduction of systematic errors in other loading conditions to no higher than 1.5% of the magnitude of the predicted stresses, far less that errors obtained under isotropic hardening, and at a level comparable to experimental uncertainty in the stress measurement. The model is expected to lead to a significant improvement in stress prediction under conditions dominated by proportional loading, and this is expected to directly improve the accuracy of springback, tearing, and earing predictions for these processes. In addition, it is shown that there is no consequence on MK necking localization due to the saturation of the yield surface in pure shear that occurs with the aluminum alloys using the present model.

Towards variational constitutive updates for non-associative plasticity models at finite strain: Models based on a volumetric-deviatoric split

In this paper, an enhanced variational constitutive update suitable for a class of non-associative plasticity theories at finite strain is proposed. In line with classical numerical formulations for plasticity models, such as the by now established return-mapping algorithm, variational constitutive updates represent a numerical method for computing the unknown state variables. However, in contrast to conventional algorithms, variational constitutive updates are fully variational, i.e., all unknown variables follow jointly from minimizing a certain potential. In addition to the physical and mathematical elegance of these variational schemes, they show several practical advantages as well. For instance, numerically efficient and robust optimization schemes can be directly employed for solving the resulting minimization problem. Since mathematically, plasticity is a non-smooth problem and often, it leads to highly singular systems of equations as known from single crystal plasticity, a robust implementation is of utmost importance. So far, variational constitutive updates have been developed for different classes of standard dissipative solids, i.e., solids characterized by associative evolution equations and flow rules. In the present paper, this framework is extended to a certain class of non-associative plasticity models at finite strain. All models falling into this class show a volumetric-deviatoric split of the Helmholtz energy and the yield function. Typical prototypes are Drucker-Prager or Mohr-Coulomb models playing an important role in soil mechanics. The efficiency and robustness of the resulting algorithmic formulation is demonstrated by means of selected numerical examples.

Multi-level Elastoplastic Damage Mechanics for Elliptical Fiber-reinforced Composites with Evolutionary Fiber Debonding

A micromechanical multi-level elastoplastic evolutionary damage framework is proposed to predict the overall mechanical behavior and interfacial damage evolutions of elliptical fiber-reinforced ductile composites. Progressively debonded fibers are replaced by equivalent microvoids. The effective elastic moduli of three phase composites, composed of a ductile matrix, randomly located yet monotonically aligned elliptical fibers and elliptical microvoids, are derived by using a micromechanical formulation. In order to characterize the homogenized elastoplastic behavior, an effective yield criterion is derived based on the ensemble-area averaging process and the first-order effects of eigenstrains. The resulting effective yield criterion, together with the overall associative plastic flow rule and the hardening law, constitutes the analytical framework for the estimation of effective elastoplastic damage responses of ductile composites containing both perfectly bonded and completely debonded fibers. An evolutionary interfacial fiber debonding process, governed by the internal stresses of fibers and the interfacial strength, is incorporated into the proposed work. The Weibull's probabilistic distribution is employed to describe the varying probability of fiber debonding. Further, systematic numerical simulations are presented to illustrate the potential of the proposed methodology.

An Integration Algorithm for Constitutive Models with Two Intersecting Yield Surfaces

: In this paper, the constitutive models involving two active intersecting yield surfaces, such as Lade's model, are considered and a new integration algorithm is formulated. The formulation introduces a new class of the return mapping algorithm within the framework of the operator splitting methodology. The proposed algorithm is applicable to a general class of elastoplastic constitutive models exhibiting associative or nonassociative flow rule, arbitrary yield criteria and hardening laws. The algorithm can handle one or two active yield surfaces at a time. The development leads to a relatively simple, straightforward algorithm for updating the stresses at Gauss point. Finally, the new algorithm is implemented in a generalized nonlinear finite element computer program incorporating Lade's model. The results of a number of boundary value problems are presented and compared with available experimental data and theoretical formulas. The comparisons indicate that incorporating the new class of return mapping algorithm gives reasonable solutions.

On the prediction of the collapse load of circular concrete columns confined by FRP

The present work is aimed at deriving assessment and design formulae for determining the elastic–plastic response and the ultimate compressive strength of circular concrete columns confined by Fiber Reinforced Polymers (FRP). To achieve this, a constructive method for obtaining closed-form elastic and post-elastic solutions for Functionally Graded Material Cylinders (FGMCs), constituted by an isotropic central core and n arbitrary cylindrically orthotropic hollow phases, is proposed. In the first part of the paper, under the hypotheses of axis-symmetrical boundary conditions, elasticity and perfect bond between the phases, new analytical solutions for self-equilibrated axial forces applied at the extremities of the object are derived. In particular, the rather general mathematical approach has been based on a strategy already proposed by some of the authors in a previous work, and here extended to anisotropic hollow phases. The key of the involved method is to reduce the differential Boundary Value Problem (BVP) to the equivalent linear algebraic one, by means of a special matrix-like arrangement of the governing equations and invoking the Complex Potential Theory for anisotropic materials. The obtained general solution has been then easily particularized to the two phase FGMC representing the circular concrete column confined by FRP sheets. In the second part of the paper, the above mentioned solutions for the concrete column are “moved” within the post-elastic range and we investigate the evolution of the stress field in the solid components when the concrete core is characterized by an Intrinsic Curve or Schleicher-like elastic–plastic behavior endowed with associate flow rule, and the FRP cylindrically orthotropic hollow phases obey to an elastic–brittle Tsai–Hill anisotropic yield criterion. At the end, the elastic and post-elastic response of the overall solid and predictive formulae for estimating the failure mechanism, in terms of concrete ultimate compressive strength, confining pressure and strain at failure, are derived. The obtained results are finally compared with several experimental literature data, highlighting the very good agreement between the analytical predictions and experimental tests.

On a numerical implementation of a formulation of anisotropic continuum elastoplasticity at finite strains

In a recent theoretical study [see C. Sansour, I. Karšaj, J. Sorić, A formulation of anisotropic continuum elastoplasticity at finite strains. Part I: Modelling, International Journal of Plasticity 22 (2006) 2346–2365], a constitutive model for anisotropic elastoplasticity at finite strains has been developed. The model is based on the multiplicative decomposition of the deformation gradient. The stored energy function as well as the flow rule has been considered as quadratic functions of their arguments. In both cases, the list of arguments is extended to include structural tensors which describe the anisotropy of the material response at hand. Non-linear isotropic hardening is considered as well. In this paper, the integration of the constitutive law is presented. The associative flow rule is integrated using the exponential map which preserves the plastic incompressibility condition. The numerical treatment of the problem is fully developed and expressions related to the local iteration and the consistent tangent operator are considered in detail. It is shown that while the consistent linearisation of the model is quite complicated, it still can be achieved if various intriguing implicit dependencies are identified and correctly dealt with. Various numerical examples of three-dimensional deformations of whole structural components are presented. The examples clearly illustrate the influence of anisotropy on finite elastoplastic deformations.

LOW VELOCITY IMPACT OF PLATE AND HEMISPHERIC IMPACTOR: PHYSICAL EXPERIMENT, NUMERICAL COMPUTATION, AND APPLICATION TO REAL SYSTEM

Steel plates subjected to low velocity impact were numerically analyzed, and physically tested. The impactor has hemispheric tip nose, and the square plate has four fixed edges. Mindlin plate theory and Hertz contact law were used for the formulation of the impact problem, and some numerical treatments for the reduced integration, Newmark time integration scheme, associative flow rule, and plastic hardening were appropriately applied. A simple and efficient analysis program for plate impact was generated, and the results were compared with those of physical impact experiments. A drop test system was installed, and the impact behavior was measured with sensors. The comparison showed reasonable results, and the main parameters of plate impact were also analyzed.

Experimental and FEM-supported investigation of wet ceramic clay extrusion for the determination of stress distributions on the applied tools’ surfaces

Aiming at assessing the stress distribution on the surface of a die mandrel during the extrusion of ceramic products, the flow of wet ground clay was simulated both in a ram extrusion device and by a FEM-based model, considering the von Mises criterion for the flow stress, the associative flow rule and the rigid–viscoplastic constitutive equation. The friction between clay and die is approached by the Tresca boundary condition, which proves a more realistic approach than the Coulomb friction law for the contact conditions between a plastically deforming material and a rigid surface. The Tresca friction factor and the material parameters of the constitutive model are determined through comparison of experimental and theoretical results. The distributions of clay sliding velocity and normal and frictional stress on the mandrel surface are then assessed through the numerical simulation model. It is found that no sticking areas appear on the mandrel surface.

Displacements of Multiblock Geotechnical Structures Subjected to Seismic Excitation

A method is presented for calculations of irreversible displacements of multiblock structures subjected to seismic excitation. Use is made of the kinematic approach of limit analysis. To make the analysis tractable a hodograph representing distribution of accelerations in the structure is introduced. Distinction is made between soils conforming to the associative and nonassociative flow rules, and the importance of this distinction is demonstrated. The yield acceleration calculated for slopes comprised of soils conforming to the nonassociative flow rule is lower than that for a soil with the same strength parameters, but its deformation governed by the associative flow rule. Consequently, the displacements predicted for the former are larger. An example of a slope is demonstrated, but the method presented is applicable to other structures, such as retaining walls and embankments.

Vertical bearing capacity for ring footings

This paper utilizes a finite difference code FLAC to study bearing capacity factor N γ for ring footings in cohesionless, frictional and ponderable soil. The present soil model employs Mohr–Coulomb yield criterion and associative flow rule. The value of N γ is found to decrease significantly with an increase in r i /r o, which is the ratio of internal radius to external radius of the ring. The value of N γ for a rough ring footing, especially for lager values of φ, is obviously higher than that for a smooth footing. The results are compared with those available in the literature.