dispersion analysis and element free galerkin solutions of second and fourth order gradient enhanced damage models gradient dependent damage formulations incorporate higher order derivatives of state variables in the constitutive equations different formulations have been derived for this gradient enhancement comparison of which is difficult in a finite element context due to higher order continuity requirements for certain formulations on the other hand the higher order continuity requirements are met naturally by element free galerkin efg shape functions thus the efg method provides a suitable tool for the assessment of gradient enhanced continuum models dispersion analyses have been carried out to compare different gradient enhanced models with the non local damage model the formulation of the additional boundary conditions is addressed numerical examples show the objectivity with respect to the discretization and the differences between various gradient formulations with second and fourth order derivatives it is shown that with the same underlying internal length scale very different results can be obtained copyright 2000 john wiley sons ltd

permutable indistinguishability operators perfect vague groups and fuzzy subgroups permutability between t indistinguishability operators is a very interesting property that is related to the compatibility of the operators with algebraic structures it will be shown that the sup t product email protected f of two t indistinguishability operators is also a t indistinguishability operator if and only if e and f are permutable t indistinguishability operators i e email protected email protected e this property will be related to the study of fuzzy subgroups fuzzy normal subgroups and vague groups the aggregation of fuzzy subgroups will also be analyzed

a damage model with evolving nonlocal interactions we present a damage model for softening materials with evolving nonlocal interactions the thermodynamic implications and the material stability issue are addressed the proposed nonlocal averaging scheme provides the obtained constitutive models with an evolving nonlocal interaction which is activated only when damage occurs in the analysis of structures made of quasi brittle materials this feature helps not only to overcome some issues with the incorrect initiation of damage but also to better control the evolving size of the active fracture process zone this is an essential feature that is usually not considered in depth in many existing nonlocal approaches to the continuum modelling of quasi brittle fracture numerical examples are given to demonstrate features of the proposed modelling approach

aggregation of fuzzy subgroups abstract given an aggregation function and two fuzzy subgroups of a group this study investigates whether the aggregation of them is also a fuzzy subgroup it is proven that if the cardinal of the group is a prime power then it is always a fuzzy subgroup for a general group it is proven that the existence of a binary relation between the fuzzy subgroups determines if their aggregation is a fuzzy subgroup moreover given two fuzzy subgroups if the aggregation of them is a fuzzy subgroup for some strictly monotone aggregation function then it is a fuzzy subgroup for any aggregation function the present study also examines conditions on the aggregation function that characterise the case where the aggregation of two arbitrary fuzzy subgroups is also a fuzzy subgroup

nonlocal models for damage and fracture comparison of approaches abstract the paper analyzes nonlocal constitutive models used in simulations of damage and fracture processes of quasibrittle materials a number of nonlocal formulations found in the literature are classified according to the type of variable subjected to nonlocal averaging analytical and numerical solutions of a simple one dimensional localization problem are presented it is shown that some of the formulations inevitably lead to residual stresses even at very late stages of the deformation process and consequently they are not capable of modeling complete separation in a widely open macroscopic crack the mechanisms leading to this specific type of stress locking are explained based on a theoretical analysis of the nonlocal constitutive equations it is also pointed out that the nonlocal approach distorts the shape of the stress strain diagram which has to be taken into account when designing an appropriate local softening law

mixed stabilized finite element methods in nonlinear solid mechanics part i formulation abstract this paper exploits the concept of stabilized finite element methods to formulate stable mixed stress displacement and strain displacement finite elements for the solution of nonlinear solid mechanics problems the different assumptions and approximations used to derive the methods are exposed the proposed procedure is very general applicable to 2d and 3d problems implementation and computational aspects are also discussed showing that a robust application of the proposed formulation is feasible numerical examples show that the results obtained compare favorably with those obtained with the corresponding irreducible formulation

gradient enhanced damage for quasi brittle materials summary conventional continuum damage descriptions of material degeneration suffer from loss of well posedness beyond a certain level of accumulated damage as a consequence numerical solutions are obtained which are unacceptable from a physical point of view the introduction of higher order deformation gradients in the constitutive model is demonstrated to be an adequate remedy to this deficiency of standard damage models a consistent numerical solution procedure of the governing partial differential equations is presented which is shown to be capable of properly simulating localization phenomena

finite element analysis of bifurcation in nonlocal strain softening solids progressive damage in brittle heterogeneous materials produces at the macroscopic level strain softening from which theoretical difficulties arise e g ill posedness and multiple bifurcation points this characteristics behavior favours spurious strain localization in numerical analyses and calls for the implementation of localization limiters for instance nonlocal damage constitutive relations the issue of possible stable or unstable equilibrium paths multiple localization zones and of the detection of bifurcation points has however never been addressed in the context of nonlocal constitutive laws we extend here the eigenmode analysis and perturbation method proposed by de borst to the study of the bifurcation and post bifurcation response of discrete nonlocal strain softening solids numerical applications on beams show that bifurcation and instability may occur in the post peak regime as opposed to the case of local constitutive relations the number of possible solutions at a bifurcation point is restricted due to the constraint introduced by the localization limiter and these solutions are shown to be mesh independent

iterative lcp solvers for non local loading unloading conditions this paper deals with the finite element analysis of a certain class of non local dissipative constitutive models where the canonical pointwise backward euler scheme cannot be employed for satisfying the loading unloading conditions in the presence of a non local dissipation the admissibility conditions in a point depend on the inelastic strain increment of the surrounding points and can be cast as a linear complementarity problem lcp involving all gauss points of the process zone in order to actually solve the lcp the use of iterative algorithms that can be easily embodied into existing fe codes is discussed the performance of the proposed algorithms is tested in 1d and 2d examples for both elastoplastic and damaging materials copyright 2003 john wiley sons ltd

damage growth modeling using the thick level set tls approach efficient discretization for quasi static loadings the thick level set tls model is a damage model containing a non local treatment that prevents from spurious localization issues it also offers an automatic transition from damage to fracture the tls approach to model damage growth was first presented in moes et al 24 with a first numerical implementation for a time dependent damage evolution law in this paper we propose some improvements in terms of discretization and explicit damage growth algorithms to obtain a robust efficient and easy to implement model these improvements include a simple and efficient variational formulation for computing the non local quantities as well as the introduction of a so called ramped heaviside enrichment function to properly take into account the transition to cracks in fully damaged zones we consider here a simple explicit formulation for quasi static loadings decoupling the elastic computation and the damage growth the method is validated through standard benchmarks and compared to the griffith s fracture theory the convergence of energy and displacement errors is observed more complex computations including damage initiations are eventually performed

a strain difference based nonlocal elasticity model abstract a two component local nonlocal constitutive model for macroscopically inhomogeneous linear elastic materials but constant internal length is proposed in which the stress is the sum of the local stress and a nonlocal type stress expressed in terms of the strain difference field hence identically vanishing in the case of uniform strain attention is focused upon the particular case of piecewise homogeneous material the proposed model is thermodynamically consistent with a suitable free energy potential it constitutes an improved form of the vermeer and brinkgreve a new effective nonlocal strain measure for softening plasticity in chambon r desrues j vardulakis i eds localization and bifurcation theory for soils and rocks balkema rotterdam 1994 pp 89 100 model and can also be considered derivable from the eringen nonlocal elasticity model through a suitable enhancement technique based on the concept of redistribution of the local stress the concept of equivalent distance is introduced to macroscopically account for the further attenuation effects produced by the inhomogeneity upon the long distance interaction forces with the aid of a piecewise homogeneous bar in tension a portion of which degrades progressively till failure it is shown that under a suitable choice of a material constant the solution procedure exhibits no pathological features numerical instability mesh sensitivity in every degraded bar condition including the limit idealized stress free condition of the failed bar

size effect and localization in j2 plasticity this paper studies the phenomenon of structural size effect and strain localization in j2 plasticity size effect is here understood as the change in the response of a given structure when the spatial dimensions are scaled up or down while the geometry and other relevant properties of the structure are preserved the work exploits the advantages of the mixed displacement pressure formulation in incompressible or quasi incompressible situations elasto j2 plastic constitutive behaviour with regularized softening is considered stability issues are discussed to ensure existence and uniqueness of the solution of the corresponding discrete finite element formulation numerical examples show that the formulation derived is able to solve a wide range of structural scales including real life engineering applications the results obtained do not suffer from spurious mesh dependence furthermore the formulation includes the classical theories of perfect plasticity and linear fracture mechanics as limit cases

a symmetric nonlocal damage theory abstract the paper presents a thermodynamically consistent formulation for nonlocal damage models nonlocal models have been recognized as a theoretically clean and computationally efficient approach to overcome the shortcomings arising in continuum media with softening the main features of the presented formulation are i relations derived by the free energy potential fully complying with nonlocal thermodynamic principles ii nonlocal integral operator which is self adjoint at every point of the solid including zones near to the solid s boundary iii capacity of regularizing the softening ill posed continuum problem restoring a meaningful nonlocal boundary value problem in the present approach the nonlocal integral operator is applied consistently to the damage variable and to its thermodynamic conjugate force i e nonlocality is restricted to internal variables only the present model when associative nonlocal damage flow rules are assumed allows the derivation of the continuum tangent moduli tensor and the consistent tangent stiffness matrix which are symmetric the formulation has been compared with other available nonlocal damage theories finally the theory has been implemented in a finite element program and the numerical results obtained for 1 d and 2 d problems show its capability to reproduce in every circumstance a physical meaningful solution and fully mesh independent results

three player absorbing games an n player absorbing game is an n player stochastic game where all the states but one are absorbing a state is absorbing if once it is reached the probability to leave it is zero whatever the players play we prove that every three player absorbing game has an undiscounted equilibrium payoff

non local damage model based on displacement averaging continuum damage models describe the changes of material stiffness and strength caused by the evolution of defects in the framework of continuum mechanics in many materials a fast evolution of defects leads to stress strain laws with softening which creates serious mathematical and numerical problems to regularize the model behaviour various generalized continuum theories have been proposed integral type non local damage models are often based on weighted spatial averaging of a strain like quantity this paper explores an alternative formulation with averaging of the displacement field damage is assumed to be driven by the symmetric gradient of the non local displacements it is demonstrated that an exact equivalence between strain and displacement averaging can be achieved only in an unbounded medium around physical boundaries of the analysed body both formulations differ and the non local displacement model generates spurious damage in the boundary layers the paper shows that this undesirable effect can be suppressed by an appropriate adjustment of the non local weight function alternatively an implicit gradient formulation could be used issues of algorithmic implementation computational efficiency and smoothness of the resolved stress fields are discussed copyright 2005 john wiley sons ltd

error assessment and mesh adaptivity for regularized continuous failure models this paper deals with the adaptive finite element analysis of structural failure a gradient enhanced damage model has been chosen to simulate material degradation since this model is regularized in the post peak regime the finite element solution does not suffer from pathological mesh dependence and thus converges to an objective solution upon mesh refinement however the error analyses have shown that the error in the nonlocal equivalent strain field becomes dominant during the post peak loading stages the accuracy of the nonlocal equivalent strain field and the corresponding damage quantity also greatly influences the accuracy of the quantity of interest two error measures have been proposed the goal oriented error estimates have provided similar error distributions although some small differences have been found in the softening regime objective error estimates together with adaptive criteria have been used to perform automated h adaptivity during computation

a general framework for softening regularisation based on gradient elasticity a general non local approach to regularise strain softening continua is presented the key idea is to introduce the gradient type enrichment at the level of displacements rather than some internal variable so the model is formulated with two distinct displacement fields in fact gradient models based on two displacement fields are usual in non local elasticity where the goal is to avoid the shortcomings of classical local elasticity i e strain singularities in statics non dispersive behaviour in dynamics we show that such a gradient elasticity backbone model can be combined with any standard nonlinear constitutive driver to render a regularised model for softening inelasticity to illustrate the generality of the approach two prototype models isotropic damage and von mises plasticity are discussed the numerical examples show that the regularised models exhibit all of the desired features mesh insensitivity imperfection size insensitivity and description of size effects

fuzzy groups fuzzy functions and fuzzy equivalence relations to each fuzzy subgroup a fuzzy equivalence relation is associated and it is proved that a fuzzy subgroup is normal if and only if the operation of the group is compatible with its associated fuzzy equivalence relation in the definition of fuzzy subgroup only the subset is fuzzy whilst the group operation remains crisp in an opposite direction another structure can be constructed maintaining crisp the basic set x of the group and fuzzifying the group operation when in addition it is imposed to the fuzzy operation to be compatible with a given fuzzy equivalence relation this structure is called a vague group the results of the paper allow us to associate a vague group to every fuzzy subgroup in such a way that it can be interpreted as the fuzzy quotient group x special attention is paid to normal fuzzy groups and vague groups of the additive group r of the real numbers two examples related to triangular numbers illustrating the results of the paper are provided

selection of a correlated equilibrium in markov stopping games abstract this paper deals with an extension of the concept of correlated strategies to markov stopping games the nash equilibrium approach to solving nonzero sum stopping games may give multiple solutions an arbitrator can suggest to each player the decision to be applied at each stage based on a joint distribution over the players decisions according to some optimality criterion this is a form of equilibrium selection examples of correlated equilibria in nonzero sum games related to the best choice problem are given several concepts of criteria for selecting a correlated equilibrium are used

a nonlocal model with strain based damage a thermodynamically consistent formulation of nonlocal damage in the framework of the internal variable theories of inelastic behaviours of associative type is presented the damage behaviour is defined in the strain space and the effective stress turns out to be additively splitted in the actual stress and in the nonlocal counterpart of the relaxation stress related to damage phenomena an important advantage of models with strain based loading functions and explicit damage evolution laws is that the stress corresponding to a given strain can be evaluated directly without any need for solving a nonlinear system of equations a mixed nonlocal variational formulation in the complete set of state variables is presented and is specialized to a mixed two field variational formulation hence a finite element procedure for the analysis of the elastic model with nonlocal damage is established on the basis of the proposed two field variational formulation two examples concerning a one dimensional bar in simple tension and a two dimensional notched plate are addressed no mesh dependence or boundary effects are apparent

a posteriori error estimation for standard finite element analysis a new residual type estimator based on projections of the error on subspaces of locally supported functions is presented the estimator is defined by a standard element by element refinement first an approximation of the energy norm of the error is obtained solving local problems with homogeneous dirichlet boundary conditions a later enrichment of the estimation is performed by adding the contributions of projections on a new family of subspaces this estimate is a lower bound of the measure of the actual error the estimator does not need to approximate local boundary conditions for the error equation therefore computation of flux jumps is not necessary moreover the estimator can be applied in mixed meshes containing elements of different shapes and its implementation in a standard finite element code is straightforward the presented results show the effectiveness of the estimator approximating both the distribution and the global measure of the error as well as its usefulness in adaptive procedures

adaptive technique for discrete models of fracture abstract static discrete models are advantageously used for the simulation of fracture in quasibrittle heterogeneous materials in order to correctly capture strain localization during the fracture process it is often necessary to represent material heterogeneity in the model directly via its discrete geometry depending on the specimen size and the size of the heterogeneities these simulations are typically extremely computationally demanding the contribution aims to reduce this computational cost via the implementation of adaptivity in the construction of the discrete model geometry the simulation starts with coarse discretization which provides correct elastic behavior and is then adaptively refined during the simulation in regions that suffer high stresses that induce cracking and strain localization the technique is applied in deterministic and probabilistic simulations and demonstrated on several examples

an equivalent stress gradient regularization model for coupled damage viscoelasticity abstract for materials that display viscoelastic behavior adequate description of their failure requires accurate prediction of damage initiation propagation and growth rate in addition to their time dependent response while many local damage models are available in the literature they all lack a length scale needed to regularize the solution and lead to mesh independent results in this work we propose a new damage regularization approach based on an equivalent stress measure concept and apply it to a prony series type viscoelastic solid with a murakami type damage rate law viscoelastic behavior is achieved by a semi analytical integration of the constitutive law and damage regularization is obtained by solving an additional second order gradient equation of an equivalent stress the scheme leads to a coupled set of nonlinear equations which are solved simultaneously using a monolithic newton framework to obtain displacement and damage fields as a function of time the formulation is shown to be thermodynamically consistent and the energy dissipation of the model is studied mesh insensitive behavior under creep relaxation and strain rate conditions is demonstrated for one and two dimensional problems moreover for different viscoelastic materials with different dominant damage terms the proposed model is shown to provide consistent damage growth results

efficient and reliable nonlocal damage models we present an efficient and reliable approach for the numerical modelling of failure with nonlocal damage models the two major numerical challenges the strongly nonlinear highly localized and parameter dependent structural response of quasi brittle materials and the interaction between non adjacent finite elements associated to nonlocality are addressed in detail efficiency is achieved with a suitable combination of load stepping control technique and nonlinear solver for equilibrium equations reliability of the numerical results is ensured by an h adaptive strategy based on error estimation we use a residual type error estimator for nonlinear fe analysis based on local computations which at the same time accounts for the nonlocality of the damage model the proposed approach is illustrated by means of three application examples the three point bending test the single edge notched beam test and the brazilian test

dispersion and isogeometric analyses of second order and fourth order implicit gradient enhanced plasticity models implicit gradient plasticity models incorporate higher order spatial gradients via an additional helmholtz type equation for the plastic multiplier so far the enrichment has been limited to second order spatial gradients resulting in a formulation that can be discretised using c0 continuous finite elements herein an implicit gradient plasticity model is formulated that includes a fourth order gradient term as well a comparison between the localisation properties of both the implicit gradient plasticity formulations and the explicit second order gradient plasticity model is made using a dispersion analysis the higher order continuity requirement for the fourth order implicit gradient plasticity model has been met by exploiting the higher order continuity property of isogeometric analysis which uses nonuniform rational b splines as shape functions instead of lagrange polynomials the discretised variables displacements and plastic multiplier may require different orders of interpolation an issue that is also addressed numerical results show that both formulations can be used as a localisation limiter but that quantitative differences occur and a different evolution of the localisation band is obtained for 2 dimensional problems

a unified approach to remeshing strategies for finite element h adaptivity in h adaptivity a remeshing strategy is needed to compute the distribution of required element size using the estimated error distribution several authors have introduced different remeshing strategies yielding very different results in this work these methods are included in a unified framework emphasizing the role of the underlying hypotheses moreover an objective tool to evaluate the accuracy of the resulting finite element solution is presented thus a new remeshing strategy is introduced to optimize the accuracy of the adapted solutions the different remeshing strategies are compared with well known numerical examples

strain localization and bifurcation in a nonlocal continuum abstract the conditions for localization and wave propagation in a strain softening material described by a nonlocal damage based constitutive relation are derived in closed form localization is understood as a bifurcation into a harmonic mode the criterion for bifurcation is reduced to the classical form of singularity of a pseudo acoustic tensor this tensor is not a material property as it involves the wavelength of the bifurcation mode through the fourier transform of the weight function used in the definition of the nonlocal damage a geometrical solution is provided to analyse localization the conditions for the onset of bifurcation are found to coincide in the nonlocal and in the corresponding local cases in the nonlocal continuum the wavelength of the localization mode is constrained to remain below a threshold which is proportional to the characteristic length of the continuum the analysis in dynamics exhibits the well known property of wave dispersion in some instances i e for large wavelength modes wave celerities become imaginary but waves with a sufficiently short wavelength are found to propagate during softening in all the situations

a topological approach to quitting games this paper presents a question of topological dynamics and demonstrates that its affirmation would establish the existence of approximate equilibria in all quitting games with only normal players a quitting game is an undiscounted stochastic game with finitely many players where every player has only two moves to end the game with certainty or to allow the game to continue if nobody ever acts to end the game all players receive payoffs of 0 a player is normal if and only if by quitting alone she receives at least her min max payoff this proof is based on a version of the kohlberg mertens kohlberg e j f mertens 1986 on the strategic stability of equilibria econometrica 54 5 1003 1037 structure theorem designed specifically for quitting games

stress based nonlocal damage model abstract progressive microcracking in brittle or quasi brittle materials as described by damage models presents a softening behavior that in turn requires the use of regularization methods in order to maintain objective results such regularization methods which describe interactions between points provide some general properties including objectivity and the non alteration of a uniform field as well as drawbacks damage initiation free boundary a modification of the nonlocal integral regularization method that takes the stress state into account is proposed in this contribution the orientation and intensity of nonlocal interactions are modified in accordance with the stress state the fundamental framework of the original nonlocal method has been retained making it possible to maintain the method s advantages the modification is introduced through the weight function which in this modified version depends not only on the distance between two points as for the original model but also on the stress state at the remote point the efficiency of this novel approach is illustrated using several examples the proposed modification improves the numerical solution of problems observed in numerical simulations involving regularization techniques damage initiation and propagation in mode i as well as shear band formation are analyzed herein

a characterization of fuzzy subgroups abstract an ordinary subgroup of a group g is 1 a subset of g 2 closed under the group operation in a fuzzy subgroup it is precisely these two notions that lose their deterministic character a fuzzy subgroup of a group g associates with each group element a number the larger the number the more certainly that element belongs to the fuzzy subgroup the closure property is captured by the inequality x y t x y in a rosenfeld s original definition t was the function minimum however any t norm t provides a meaningful generalization of the closure property two classes of fuzzy subgroups are investigated the fuzzy subgroups in one class are subgroup generated those in the other are function generated each fuzzy subgroup in these classes satisfies the above inequality with t given by t a b max a b 1 0 while the two classes look different each fuzzy subgroup in either is isomorphic to one in the other it is shown that a fuzzy subgroup satisfies the above inequality with t minimum if and only if it is subgroup generated of a very special type finally these notions are applied to some abstract pattern recognition problems

convergence adaptive refinement and scaling in 1d peridynamics we introduce here adaptive refinement algorithms for the non local method peridynamics which was proposed in j mech phys solids 2000 48 175 209 as a reformulation of classical elasticity for discontinuities and long range forces we use scaling of the micromodulus and horizon and discuss the particular features of adaptivity in peridynamics for which multiscale modeling and grid refinement are closely connected we discuss three types of numerical convergence for peridynamics and obtain uniform convergence to the classical solutions of static and dynamic elasticity problems in 1d in the limit of the horizon going to zero continuous micromoduli lead to optimal rates of convergence independent of the grid used while discontinuous micromoduli produce optimal rates of convergence only for uniform grids examples for static and dynamic elasticity problems in 1d are shown the relative error for the static and dynamic solutions obtained using adaptive refinement are significantly lower than those obtained using uniform refinement for the same number of nodes copyright 2008 john wiley sons ltd

preserving fuzzy subgroups and indistinguishability operators abstract in this paper we study the class of all fuzzy subgroups defined with respect to a given t norm on a fixed group we find necessary and sufficient conditions over two t norms to guarantee that the class of all fuzzy subgroups induced by the first t norm is contained in the class of all fuzzy subgroups with respect to by the second one we characterize the functions that transform fuzzy subgroups into fuzzy subgroups the close relationship between indistinguishability operators and fuzzy subgroups allows us to obtain similar results for some classes of indistinguishability operators

nonlocal elasticity and related variational principles abstract the eringen model of nonlocal elasticity is considered and its implications in solid mechanics studied the model is refined by assuming an attenuation function depending on the geodetical distance between material particles such that in the diffusion processes of the nonlocality effects certain obstacles as holes or cracks existing in the domain can be circumvented a suitable thermodynamic framework with nonlocality is also envisaged as a firm basis of the model the nonlocal elasticity boundary value problem for infinitesimal displacements and quasi static loads is addressed and the conditions for the solution uniqueness are established three variational principles nonlocal counterparts of classical ones i e the total potential energy the complementary energy and the mixed hu washizu principles are provided the former is used to derive a nonlocal type fem nl fem in which the symmetric global stiffness matrix reflects the nonlocality features of the problem an alternative standard fem based solution method is also provided which consists in an iterative procedure of the type local prediction nonlocal correction in which the nonlocality is simulated by an imposed like correction strain the potentialities of these analysis methods are pointed out their numerical implementations being the object of an ongoing research work

mixed stabilized finite element methods in nonlinear solid mechanics part ii strain localization this paper deals with the question of strain localization associated with materials which exhibit softening due to tensile straining a standard local isotropic rankine damage model with strain softening is used as exemplary constitutive model both the irreducible and mixed forms of the problem are examined and stability and solvability conditions are discussed lack of uniqueness and convergence difficulties related to the strong material nonlinearities involved are also treated from this analysis the issue of local discretization error in the pre localization regime is deemed as the main difficulty to be overcome in the discrete problem focus is placed on low order finite elements with continuous strain and displacement fields triangular p1p1 and quadrilateral q1q1 although the presented approach is very general numerical examples show that the resulting procedure is remarkably robust it does not require the use of auxiliary tracking techniques and the results obtained do not suffer from spurious mesh bias dependence

a plastic nonlocal damage model abstract in the present paper a plastic nonlocal damage model is proposed for studying the mechanical response of structural elements made of cementitious materials a new isotropic damage model which is able to describe the behavior of a wide class of cementitious materials is presented a regularization technique based on the introduction of the damage laplacian in the damage limit function is adopted to overcome the analytical and computational problems induced by the softening constitutive law a drucker prager type of plastic limit function is proposed considering isotropic hardening a numerical procedure based on an implicit backward euler technique for the time integration of the plastic and damage evolution equations is presented to solve each nonlinear step a predictor corrector iterative method is developed within the splitting method in particular the damage evolution is determined solving a constrained minimization problem of a convex functional the proposed algorithm is implemented in a finite element code and it is used to study the structural behavior of elements made of masonry materials

permutable fuzzy consequence and interior operators and their connection with fuzzy relations fuzzy operators are an essential tool in many fields and the operation of composition is often needed in general composition is not a commutative operation however it is very useful to have operators for which the order of composition does not affect the result in this paper we analyze when permutability appears that is when the order of application of the operators does not change the outcome we characterize permutability in the case of the composition of fuzzy consequence operators and the dual case of fuzzy interior operators we prove that for these cases permutability is completely connected to the preservation of the operator type we also study the particular case of fuzzy operators induced by fuzzy relations through zadeh s compositional rule and the inf composition for this cases we connect permutability of the fuzzy relations using the sup composition with permutability of the induced operators special attention is paid to the cases of operators induced by fuzzy preorders and similarities finally we use these results to relate the operator induced by the transitive closure of the composition of two reflexive fuzzy relations with the closure of the operator this composition induces

gradient dependent plasticity formulation and algorithmic aspects a plasticity theory is proposed in which the yield strength not only depends on an equivalent plastic strain measure hardening parameter but also on the laplacian thereof the consistency condition now results in a differential equation instead of an algebraic equation as in conventional plasticity to properly solve the set of non linear differential equations the plastic multiplier is discretized in addition to the usual discretization of the displacements for appropriate boundary conditions this formulation can also be derived from a variational principle accordingly the theory is complete

error estimation and adaptivity for nonlocal damage models abstract nonlocal damage models are typically used to model failure of quasi brittle materials due to brittleness the choice of a particular model or set of parameters can have a crucial influence on the structural response to assess this influence it is essential to keep finite element discretization errors under control if not the effect of these errors on the result of a computation could be erroneously interpreted from a constitutive viewpoint to ensure the quality of the fe solution an adaptive strategy based on error estimation is proposed here it is based on the combination of a residual type error estimator and quadrilateral h remeshing another important consequence of brittleness is that it leads to structural responses of the snap through or snap back type this requires the use of arc length control with a definition of the arc parameter that accounts for the localized nature of quasi brittle failure all these aspects are discussed for two particular nonlocal damage models mazars and modified von mises and for two tests the brazilian tensile splitting test and the single edge notched beam test for the latter test the capability of the mazars model to capture the curved crack pattern observed in experiments a topic of debate in the literature is confirmed

correlated equilibrium in quitting games a quitting game is a sequential game where each player has two actions to continue or to quit the game continues as long as all players decide to continue the moment at least one player decides to quit the game terminates the terminal payoff depends on the subset of players who quit at the terminating stage if the game continues forever then the payoff for the players is some fixed payoff vector we prove that every quitting game admits a correlated uniform e equilibrium a uniform e equilibrium in an extended game that includes a correlation device that sends one signal to each player before start of play

fe formulation of a nonlocal plasticity theory a nonlocal continuum plasticity theory is presented the nonlocal field introduced here is defined as a certain weighted average of the corresponding local field taken over all the material points in the body hereby a quantity with the dimension of length occurs as a material parameter when this so called internal length is equal to zero the local classical plasticity theory is regained in the present model the yield function will depend on a nonlocal field the consistency condition and the integration algorithm result in integral equations for determination of the field of plastic multipliers the integral equations are classified as fredholm equations of the second kind and the existence of a solution will be commented upon after discretization a matrix equation is obtained and an algorithm for finding the solution is proposed for a generalized von mises material a plane boundary value problem is solved with a fe method since the nonlocal quantities are integrals c0 continuous elements are sufficient the solution strategy is split into a displacement estimate for equilibrium and the integration of constitutive equations in the numerical simulations shear band formation is analysed and the results display mesh insensitivity

some properties of fuzzy topological groups abstract it is well known that if s is a subgroup of a topological group g so is s if s is a normal subgroup of g so is s and if u is the system of all neighborhoods of the identity e in a topological group x then b bu ub for any subset b of x we extend these to the corresponding theorems in fuzzy topological groups that is based on the q neighborhood system of fuzzy points and the structure of a fuzzy topological group defined by ma and yu fuzzy sets and systems 12 1984 289 we characterize some basic properties of fuzzy topological groups show that if h is a fuzzy subgroup of a fuzzy topological group x and h h h h so is h and if h is a fuzzy normal subgroup of x and h h h h so is h and show that for any fuzzy subset a of a fuzzy topological group if x p aw wa and 0 5 then x p a and if x p a and p q then aw x wa x 0 where w is the system of all q neighborhoods of the identity e q

error estimation including pollution assessment for nonlinear finite element analysis a residual type error estimator for nonlinear finite element analysis is introduced this error estimator solves local problems avoiding both the computation of the flux jumps and the associated flux splitting procedure pollution errors are taken into account by a feedback strategy that is an error estimate based on local computations is used as the input of the pollution analysis this estimator is used in the frame of an adaptive procedure numerical examples show that the estimator is able to drive adaptive procedures leading to likely good solutions moreover one of the examples demonstrates that adaptive procedures are essential for complex highly nonlinear mechanical problems because they may discover secondary collapse mechanisms

an algorithm for gradient regularized plasticity coupled to damage based on a dual mixed fe formulation abstract a thermodynamically consistent theory of gradient regularized plasticity with coupling to isotropic damage was presented by svedberg and runesson 23 its main purpose was to describe the successive development of the localization zone that is pertinent to the softening regime due to excessive damage close to failure explicit constitutive relations were established for a model problem based on von mises plasticity with isotropic hardening of two types local and gradient enhanced the main purpose of this paper is to describe a new numerical algorithm for the gradient enhanced formulation using the backward euler rule leads to a gradient enhanced format of the classical closest point projection method the plastic multiplier increment is solved via a dual mixed method which is a convenient feature in the present context this leads to an unambiguous representation of the constitutive relations in the equilibrium equation moreover the classical local algebraic kuhn tucker conditions are obtained in a straightforward fashion in the limit case of local theory the developed algorithm was implemented in the educational finite element package 5 numerical results are shown for the extension of a plate in plane stress these results demonstrate the convergence characteristics with mesh refinement as well as the salient features of the adopted damage based constitutive model