a continuum model accounting for defect and mass densities in solids with inelastic material behaviour in this paper the analysis of structures with inelastic material behaviour is considered taking into account the evolution of defects and changes in mass density the underlying kinematical concept of an oriented continuum is general enough to describe the micro and macrobehaviour of material bodies appropriately based on the logical and consistent variational arguments for a lagrangian functional the dynamic balance laws boundary and transversality conditions all related to the evolution of defect density and mass changes are derived for macro and microstresses of deformational as well as of configurational type the adopted procedure which formally leaves the balance laws unaltered leads to the additional balance law for changes in defect density and additional boundary conditions for the changes in mass and defect densities driving forces or affinities associated with the evolution of defect and mass densities and a generalization of the j integral representing the thermodynamic forces on defects are obtained a nonlocal constitutive model accounting for changes in the defect density is presented

a mixed finite element for the analysis of large inelastic strains a 4 node element with 1 integration point is developed for the analysis of elastoplastic large strains it is based on the hu washizu principle and uses a co rotational formulation special care is taken to avoid spurious zero energy modes and incompressibility locking this element successfully passes the large strain patch test it proves to be accurate and computationally very cheap

efficient modeling of localized material failure by means of a variationally consistent embedded strong discontinuity approach this paper is concerned with a novel embedded strong discontinuity approach suitable for the analysis of material failure at finite strains focus is on localized plastic deformation particularly relevant for slip bands in contrast to already existing models the proposed implementation allows to consider several interacting discontinuities in each finite element based on a proper re formulation of the kinematics an efficient parameterization of the deformation gradient is derived it permits to compute the strains explicitly that improves the performance significantly however the most important novel contribution of the present paper is the advocated variational constitutive update within this framework every aspect is naturally driven by energy minimization i e all unknown variables are jointly computed by minimizing the stress power the proposed update relies strongly on an extended principle of maximum dissipation this framework provides enough flexibility for different failure types and for a broad class of non associative evolution equations by discretizing the aforementioned continuous variational principle an efficient numerical implementation is obtained it shows in addition to its physical and mathematical elegance several practical advantages for instance the physical minimization principle itself specifies automatically and naturally the set of active strong discontinuities copyright 2011 john wiley sons ltd

finite element method for gradient plasticity at large strains a finite element method for gradient elasto plastic continuum in which the yield strength of strain hardening softening materials not only depends on the effective plastic strain but also on its laplacian is presented the consistent integration algorithm to update the stress and the internal state variable at integration points and the consistent compliance matrix for the gradient plasticity are formulated in the non local sense the methodology to derive the finite element formulation for the gradient plasticity at large strains presented in this paper is applicable to general finite element analysis the formulation in the context of the two dimensional four noded mixed finite element with one integration point and mean von mises yield criterion is particularly derived numerical examples are tested to demonstrate the capability and performance of the present finite element method at large strain in solving for the strain localization problem

gradient constitutive relations numerical aspects and application to gradient damage abstract the description of localisation and structural failure through local strain softening constitutive laws leads to ill posed problems to retrieve a well posed problem the material state is described through the gradients of internal variables in addition to the strain and the internal variables themselves under some assumptions it is shown that these gradient laws can be cast into a variational formulation after time discretisation the evolution of the internal variable field is characterised by a minimum principle a specific algorithm is proposed to perform the minimisation in the context of finite elements finally it is applied to a gradient brittle damage model

a discontinuous galerkin formulation of a model of gradient plasticity at finite strains abstract the work presented here constitutes an extension to the finite strain regime of a discontinuous galerkin based strain gradient plasticity formulation presented in djoko et al j k djoko f ebobisse a t mcbride b d reddy a discontinuous galerkin formulation for classical and gradient plasticity part 1 formulation and analysis comput methods appl mech engrg 196 2007 3881 3897 j k djoko f ebobisse a t mcbride b d reddy a discontinuous galerkin formulation for classical and gradient plasticity part 2 algorithms and numerical analysis comput methods appl mech engrg 197 2007 1 21 the focus here is on algorithmic and computational aspects of the formulation at finite strains the adoption of a logarithmic hyperelastic plastic formulation preserves the essential features of the infinitesimal formulation this key ingredient allows the predictor corrector solution algorithms developed for the infinitesimal gradient formulation to be extended readily to the finite strain regime the use of low order elements is essential to contain the computational expense of the formulation but these elements are prone to locking the method of enhanced assumed strains for geometrically nonlinear problems is utilised to circumvent this limitation the form of the consistent tangent modulus is derived for the case of gradient plasticity two numerical examples are presented to illustrate aspects of the approximation scheme and the algorithm as well as features of the model of gradient plasticity

an upper bound on the critical percolation probability for the three dimensional cubic lattice we prove that the critical probability for site percolation on the three dimensional cubic lattice satisfies the inequality p 3 c 1 2 an application to the three dimensional ising model is given

an adaptive finite element algorithm for gradient theory of plasticity with coupling to damage in this paper an adaptive finite element algorithm is proposed for resolving localization zones in plastic solids with coupling to damage whereby the solution is regularized via the introduction of gradient theory it is possible to avoid the mesh dependence that otherwise would become inevitable when the localization narrows progressively due to the development of damage the adaptive method is based on an a posteriori error computation in energy norm for the plastic multiplier based on the constitutive subproblem which is solved via a mixed finite element formulation the performance of the proposed method is investigated for a few simple numerical examples in 1d and 2d

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

an analysis of a new class of integration algorithms for elastoplastic constitutive relations an accuracy analysis of a new class of integration algorithms for finite deformation elastoplastic constitutive relations recently proposed by the authors is carried out in this paper for simplicity attention is confined to infinitesimal deformations the integration rules under consideration fall within the category of return mapping algorithms and follow in a straightforward manner from the theory of operator splitting applied to elastoplastic constitutive relations general rate independent and rate dependent behaviour with plastic hardening or softening associated or non associated flow rules and nonlinear elastic response can be efficiently treated within the present framework isoerror maps are presented which demonstrate the good accuracy properties of the algorithm even for strain increments much larger than the characteristic strains at yielding

an implicit bem formulation for gradient plasticity and localization phenomena the aim of this paper is to discuss a boundary element formulation for non linear structural problems involving localization phenomena in order to overcome the well known mesh dependency observed in local plasticity a gradient plasticity model is used an implicit boundary element formulation is proposed and the underlying consistent tangent operator defined this formulation is based on the classical displacement and strain integral representations combined with an integral representation of the plastic multiplier first numerical examples are presented to illustrate the application of the method copyright 2001 john wiley sons ltd

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

mixed finite element formulations of strain gradient elasticity problems theories on intrinsic or material length scales find applications in the modeling of size dependent phenomena in elasticity length scales enter the constitutive equations through the elastic strain energy function which in this case depends not only on the strain tensor but also on gradients of the rotation and strain tensors in the present paper the strain gradient elasticity theories developed by mindlin and co workers in the 1960s are treated in detail in such theories when the problem is formulated in terms of displacements the governing partial differential equation is of fourth order if traditional finite elements are used for the numerical solution of such problems then c 1 displacement continuity is required an alternative mixed finite element formulation is developed in which both the displacement and the displacement gradients are used as independent unknowns and their relationship is enforced in an integralsense a variational formulation is developed which can be used for both linear and non linear strain gradient elasticity theories the resulting finite elements require only c 0 continuity and are simple to formulate the proposed technique is applied to a number of problems and comparisons with available exact solutions are made 2002 elsevier science b v all rights reserved

consistent tangent operators for rate independent elastoplasticity abstract it is shown that for problems involving rate constitutive equations such as rate independent elastoplasticity the notion of consistency between the tangent stiffness operator and the integration algorithm employed in the solution of the incremental problem plays a crucial role in preserving the quadratic rate of asymptotic convergence of iterative solution schemes based upon newton s method within the framework of closest point projection algorithms a methodology is presented whereby tangent operators consistent with this class of algorithms may be systematically developed to wit associative j 2 flow rules with general nonlinear kinematic and isotropic hardening rules as well as a class of non associative flow rules are considered the resulting iterative solution scheme preserves the asymptotic quadratic convergence characteristic of newton s method whereas use of the socalled elastoplastic tangent in conjunction with a radial return integration algorithm a procedure often employed results in newton type of algorithms with suboptimal rate of convergence application is made to a set of numerical examples which include saturation hardening laws of exponential type

a thermodynamically motivated implicit gradient damage framework and its application to brick masonry cracking a new gradient damage model is derived starting from thermodynamic statements nonstandard terms are added to the free energy potential for this purpose in contrast with existing approaches these terms are related to the kinematics of the problem and therefore come into play already in the elastic regime following the argument that the second law of thermodynamics need only be satisfied globally for a nonlocal continuum an energy balance equation is derived which must be solved together with the usual equilibrium equations numerical solutions of this coupled set of equations have been generated for a one dimensional problem in order to study the localisation behaviour of the new model in the limit where the elastic modulus related to the energy balance goes to 0 an earlier gradient damage model is retrieved this simplified model is used to study cracking of brick masonry under the influence of mechanical loading

gradient elasticity in statics and dynamics an overview of formulations length scale identification procedures finite element implementations and new results in this paper we discuss various formats of gradient elasticity and their performance in static and dynamic applications gradient elasticity theories provide extensions of the classical equations of elasticity with additional higher order spatial derivatives of strains stresses and or accelerations we focus on the versatile class of gradient elasticity theories whereby the higher order terms are the laplacian of the corresponding lower order terms one of the challenges of formulating gradient elasticity theories is to keep the number of additional constitutive parameters to a minimum we start with discussing the general mindlin theory that in its most general form has 903 constitutive elastic parameters but which were reduced by mindlin to three independent material length scales further simplifications are often possible in particular the aifantis theory has only one additional parameter in statics and opens up a whole new field of analytical and numerical solution procedures we also address how this can be extended to dynamics an overview of length scale identification and quantification procedures is given finite element implementations of the most commonly used versions of gradient elasticity are discussed together with the variationally consistent boundary conditions details are provided for particular formats of gradient elasticity that can be implemented with simple linear finite element shape functions new numerical results show the removal of singularities in statics and dynamics as well as the size dependent mechanical response predicted by gradient elasticity

uniqueness studies in boundary value problems involving some second gradient models abstract enhanced models such as second gradient models have been mainly theoretically developed during the sixties and the seventies new attention has been paid to these models recently when it has been recognized that classical inelastic constitutive equations especially the ones not obeying a normality rule induce ill posed boundary value problems however little work has been done to study the well posedness properties of boundary value problems involving these enhanced models the aim of this paper is a partial study of this point only uniqueness of solution of numerical discretization of boundary value problems involving second gradient models is considered we extend to second gradient models a numerical search method to find non uniqueness already established for classical media in the case of loss of uniqueness for the underlying model our numerical results show clearly that in some cases uniqueness is lost for second gradient model as well it seems that second gradient extensions of classical models do not restore the uniqueness properties but only the objectivity of the numerical computation finally our numerical experiments show once more that post peak behavior is a structural effect they also suggest that some constitutive equations often used in geomaterials modeling may be questionable and may not be complete because even adding second gradient enhancement to these models yields non unique solutions

effective equidistribution for closed orbits of semisimple groups on homogeneous spaces we prove effective equidistribution with polynomial rate for large closed orbits of semisimple groups on homogeneous spaces under certain technical restrictions notably the acting group should have finite centralizer in the ambient group the proofs make extensive use of spectral gaps and also of a closing lemma for such actions

numerical technique in plasticity including solution advancement control numerical techniques applied to the consistent formulation of plasticity which is based on convex analysis are investigated for each time step the stress is found as the projection in complementary energy of the elastic stress onto the set of plastically admissible stresses while the velocity field is the extremal of a non quadratic functional explicit formulas for von mises yield criterion with mixed hardening are developed and the nonlinear equations arising from finite element discretization are solved for comparison by a number of newton type iteration procedures with line search and are length control a few numerical examples with proportional and non proportional loading are analyzed

cohesive zone models higher order continuum theories and reliability methods for computational failure analysis a concise overview is given of various numerical methods that can be used to analyse localization and failure in engineering materials the importance of the cohesive zone approach is emphasized and various ways to incorporate the cohesive zone methodology in discretization methods are discussed numerical representations of cohesive zone models suffer from a certain mesh bias for discrete representations this is caused by the initial mesh design while for smeared representations it is rooted in the ill posedness of the rate boundary value problem that arises upon the introduction of decohesion a proper representation of the discrete character of cohesive zone formulations which avoids any mesh bias can be obtained elegantly when exploiting the partition of unity property of finite element shape functions the effectiveness of the approach is demonstrated for some examples at different scales moreover examples are shown how this concept can be used to obtain a proper transition from a plastifying or damaging continuum to a shear band with gross sliding or to a fully open crack true discontinuum when adhering to a continuum description of failure higher order continuum models must be used meshless methods are ideally suited to assess the importance of the higher order gradient terms as will be shown finally regularized strain softening models are used in finite element reliability analyses to quantify the probability of the emergence of various possible failure modes

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

damage modeling employing strain gradient continuum theory a damage model for quasi brittle materials embedded into the two dimensional c1 continuity triangular finite element formulation based on the strain gradient continuum theory is considered the isotropic damage law is applied to the higher order stress strain constitutive model which enables the analysis of both homogeneous and heterogeneous materials such softening formulation also ensures a decrease of the intensity of the material nonlocality associated with the damage growth which is necessary for the correct description of the narrow localized deformation in order to obtain the required constitutive matrices the second order homogenization procedure is applied to the various representative volume elements in the frame of a multiscale approach the derived finite element formulation is implemented into the finite element program abaqus by means of user subroutines the superior regularization capabilities as well as the accuracy and efficiency of the proposed higher order gradient damage model are demonstrated by the standard benchmark examples

two gradient plasticity theories discretized with the element free galerkin method abstract in this paper the element free galerkin method is exploited to analyze gradient plasticity theories the use of this discretization method has the advantage that higher order continuity which can be required for the plastic multiplier field is provided for in passing we show that the regularization properties of the higher order gradients are necessary since similar to finite element methods a severe discretization sensitivity is encountered otherwise the mesh free discretization method is first applied to an established stress space gradient plasticity theory next a strain space gradient plasticity theory which employs an implicit averaging of an invariant strain measure is proposed and elaborated this theory is insensitive to the order of the interpolation polynomials and as a result the convergence of the newton raphson procedure is better the performance of both gradient plasticity models is examined for the mesh free discretization scheme a comparative study is carried out for a one dimensional bar in tension while the influence of the discretization in biaxial compression is studied next including that of the numerical parameters of the discretization scheme and that of the internal length scale contained in the gradient plasticity theories

numerically consistent regularization of force based frame elements recent advances in the literature regularize the strain softening response of force based frame elements by either modifying the constitutive parameters or scaling selected integration weights although the former case maintains numerical accuracy for strain hardening behavior the regularization requires a tight coupling of the element constitutive properties and the numerical integration method in the latter case objectivity is maintained for strain softening problems however there is a lack of convergence for strain hardening response to resolve the dichotomy between strain hardening and strain softening solutions a numerically consistent regularization technique is developed for force based frame elements using interpolatory quadrature with two integration points of prescribed characteristic lengths at the element ends owing to manipulation of the integration weights at the element ends the solution of a vandermonde system of equations ensures numerical accuracy in the linear elastic range of response comparison of closed form solutions and published experimental results of reinforced concrete columns demonstrates the effect of the regularization approach on simulating the response of structural members copyright 2008 john wiley sons ltd

localization properties of strain softening gradient plasticity models part ii theories with gradients of internal variables the second part of this paper compares and evaluates enhancements of the conventional plasticity theory by gradients of internal variables attention is focused on their performance as localization limiters both explicit and implicit gradient formulations are considered it is shown that certain models suffer by serious mathematical deficiencies that would complicate their numerical implementation some other models are appropriate only at early stages of the softening process but later exhibit locking accompanied by a spurious expansion of the localized plastic zone the comparative study indicates that a convenient and robust tool for regularized modeling of the entire localization process is provided by the implicit gradient approach combined with a suitable form of the hardening softening law

numerical algorithms and mesh dependence in the disturbed state concept the disturbed state concept dsc provides a unified approach for constitutive modelling of engineering materials including such factors as elastic plastic and creep strains microcracking damage and softening and sti ening responses the interacting mechanisms in the material mixture composed of the relative intact and fully adjusted states provide implicitly for various factors such as microcrack interaction and character istic dimension the dsc model can allow for well posedness reduction or elimination of spurious mesh dependence and localization a number of problems are solved to illustrate convergence and uniqueness of the finite element procedures localization spurious mesh dependence and validation with respect to observed behavior of simulated and practical problems 1997 by john wiley sons ltd

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

higher order continua derived from discrete media continualisation aspects and boundary conditions abstract in this paper the derivation of higher order continuum models from a discrete medium is addressed with the following aims i for a given discrete model and a given coupling of discrete and continuum degrees of freedom the continuum should be defined uniquely ii the continuum is isotropic and iii boundary conditions are derived consistently with the energy functional and the equations of motion of the continuum firstly a comparison is made between two continualisation methods namely based on the equations of motion and on the energy functional they are shown to give identical results secondly the issue of isotropy is addressed a new approach is developed in which two rather than one layers of neighbouring particles are considered finally the formulation and interpretation of boundary conditions is treated by means of the hamilton ostrogradsky principle boundary conditions are derived that are consistent with the energy functional and the equations of motion a relation between standard stresses and higher order stresses is derived and used to make an interpretation of the higher order stresses an additional result of this study is the non uniqueness of the higher order contributions to the energy

equivalent plastic strain gradient enhancement of single crystal plasticity theory and numerics we propose a visco plastic strain gradient plasticity theory for single crystals the gradient enhancement is based on an equivalent plastic strain measure two physically equivalent variational settings for the problem are discussed a direct formulation and an alternative version with an additional micromorphic like field variable which is coupled to the equivalent plastic strain by a lagrange multiplier the alternative formulation implies a significant reduction of nodal degrees of freedom the local algorithm and element stiffness matrices of the finite element discretization are discussed numerical examples illustrate the advantages of the alternative formulation in three dimensional simulations of oligo crystals by means of the suggested formulation complex boundary value problems of the proposed plastic strain gradient theory can be solved numerically very efficiently

theory and numerics of a thermodynamically consistent framework for geometrically linear gradient plasticity the paper presents the theory and the numerics of a thermodynamically consistent formulation of gradient plasticity at small strains starting from the classical local continuum formulation which fails to produce physically meaningful and numerically converging results within localization computations a thermodynamically motivated gradient plasticity formulation is envisioned the model is based on an assumption for the helmholtz free energy incorporating the gradient of the internal history variable a yield condition and the postulate of maximum dissipation resulting in an associated structure as a result the driving force conjugated to the hardening evolution is identified as the quasi non local drag stress which incorporates besides the strictly local drag stress essentially the divergence of a vectorial hardening flux at the numerical side besides the balance of linear momentum the algorithmic consistency condition has to be solved in weak form thereby the crucial issue is the determination of the active constraints exhibiting plastic loading which is solved by an active set search algorithm borrowed from convex non linear programming moreover different discretization techniques are proposed in order to compare the fe performance in local plasticity with the advocated gradient formulation both for hardening and softening copyright 2001 john wiley sons ltd

spectral analysis of localization in nonlocal and over nonlocal materials with softening plasticity or damage the paper shows that spectral wave propagation analysis reveals in a simple and clear manner the effectiveness of various regularization techniques for softening materials i e materials for which the yield limits soften as a function of the total strain both plasticity and damage models are considered it is verified analytically in a simple way that the nonlocal integral type model with degrading yield limit depending on the total strain works correctly if and only one adopts an unconventional nonlocal formulation introduced in 1994 by vermeer and brinkgreve and in 1996 by planas and by stromberg and ristinmaa which is here called for the sake of brevity over nonlocal because it uses a linear combination of local and nonlocal variables in which a negative weight imposed on the local variable is com pensated by assigning to the nonlocal variable weight greater than 1 this is equivalent to a nonlocal variable with a smooth positive weight function of total weight greater than 1 normalized by superposing a negative delta function spike at the center the spectral approach readily confirms that the nonlocal integral type generalization of softening plasticity with an additive format gives correct localization properties only if an over nonlocal formulation is adopted by contrast the nonlocal integral type generalization of softening plasticity with a multiplicative format provides real istic localization behavior just like the nonlocal integral type damage model and thus does not necessitate an over non local formulation the localization behavior of explicit and implicit gradient type models is also analyzed a simple analysis shows that plasticity and damage models with gradient type localization limiter whether explicit or implicit have very different localization behaviors 2005 elsevier ltd all rights reserved

a mixed element method in gradient plasticity for pressure dependent materials and modelling of strain localization this paper presents a mixed strain element method in gradient plasticity for pressure dependent materials at large strains and its application to the modelling of strain localization the two yield strength parameters of strain hardening softening materials not only depend on the internal state variable but also on its laplacian the evaluation of the laplacian is based on a least square polynomial approximation of the internal state variable around each integration point the present non local approach allows to satisfy exactly the consistency condition in a pointwise fashion but in the non local sense and at each iteration of a load step to derive the consistent element formulations in the context of mixed finite element with one point quadrature and hourglass control and the non local pressure dependent elasto plasticity a natural coordinate system in the stress space and a new definition of internal state variable are introduced the capability of the present approach to preserve ellipticity as strain softening behaviour is incorporated into a computational model is demonstrated by the numerical results which illustrate that pathological mesh dependence has been overcome

finite strain logarithmic hyperelasto plasticity with softening a strongly non local implicit gradient framework this paper addresses the extension of a eulerian logarithmic finite strain hyperelasto plasticity model in order to incorporate an isotropic plastic damage variable that leads to softening and failure of the plastic material it is shown that a logarithmic elasto plastic model with a strongly non local degrading yield stress exactly preserves the structure of its infinitesimal counterpart the strongly non local nature of the model makes it an attractive framework for the numerical solution of softening plasticity problems consistent constitutive tangent operators are derived for the particular case of hyperelasto j2 plasticity which are exactly equal to the corresponding infinitesimal tangent operators the finite element implementation along with the geometrically nonlinear contributions and the incremental solution strategy is outlined a benchmark example is solved illustrating the main differences between the purely elasto plastic case and the case with plastic damage finally the main model characteristics and its practical use are emphasized

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

gradient elasticity and nonstandard boundary conditions abstract gradient elasticity for a second gradient model is addressed within a suitable thermodynamic framework apt to account for nonlocality the pertinent thermodynamic restrictions upon the gradient constitutive equations are derived which are shown to include besides the field differential stress strain laws a set of nonstandard boundary conditions consistently with the latter thermodynamic requirements a surface layer with membrane stresses is envisioned in the strained body which together with the above nonstandard boundary conditions make the body constitutively insulated i e no long distance energy flows out of the boundary surface due to nonlocality the total strain energy is shown to include a bulk and surface strain energy a minimum total potential energy principle is provided for the related structural boundary value problem the toupin mindlin polar type strain gradient material model is also addressed and compared with the above one their substantial differences are pointed out particularly for what regards the constitutive equations and the boundary conditions accompanying the solving displacement equilibrium equations a gradient one dimensional bar sample in tension is considered for a few applications of the proposed theory

gradient plasticity theory with a variable length scale parameter the definition and magnitude of the intrinsic length scale are keys to the development of the gradient plasticity theory that incorporates size effects however a fixed value of the material length scale is not always realistic and different problems could require different values moreover a linear coupling between the local and nonlocal terms in the gradient plasticity theory is not always realistic and that different problems could require different couplings this work addresses the proper modifications required for the full utility of the current gradient plasticity theories in solving the size effect problem it is shown that the current gradient plasticity theories do not give sound interpretations of the size effects in micro bending and micro torsion tests if a definite and fixed length scale parameter is used a generalized gradient plasticity model with a non fixed length scale parameter is proposed based on dislocation mechanics this model assesses the sensitivity of predictions to the way in which the local and nonlocal parts are coupled or to the way in which the statically stored and geometrically necessary dislocations are coupled in addition a physically based relation for the length scale parameter as a function of the course of deformation and the material microstructural features is proposed the proposed model gives good predictions of the size effect in micro bending tests of thin films and micro torsion tests of thin wires

studies in anisotropic plasticity with reference to the hill criterion algorithms based upon the notion of return mapping have been developed for the hill yield function of anisotropic plasticity the relative accuracy of two algorithms is assessed by means of iso error maps the choice of the algorithm turns out to be much more critical for the orthotropic hill criterion than for the underlying isotropic von mises plasticity model a tangent operator that is consistent with the developed integration algorithm is formulated and its efficiency is assessed compared with the classical continuum tangent operator the model has been applied to three shell plate structures

multiscale computational analysis in mechanics using finite calculus an introduction abstract the paper introduces a general procedure for computational analysis of a wide class of multiscale problems in mechanics using a finite calculus fic formulation the fic approach is based in expressing the governing equations in mechanics accepting that the domain where the standard balance laws are established has a finite size this introduces naturally additional terms into the classical equations of infinitesimal theory in mechanics which are useful for the numerical solution of problems involving different scales in the physical parameters the discrete nodal values obtained with the fic formulation and the finite element method fem can be effectively used as the starting point for obtaining a more refined solution in zones where high gradients of the relevant variables occur using hierarchical or enriched fem typical multiscale problems in mechanics which can be solved with the fic method include convection diffusion reaction problems with high localized gradients incompressible problems in solid and fluid mechanics localization problems such as prediction of shear bands in solids and shock waves in compressible fluids turbulence etc the paper presents an introduction of the treatment of multiscale problems using the fic approach in conjunction with the fem examples of application of the fic fem formulation to the solution of simple multiscale convection diffusion problems are given

variational principles for softening gradient dependent plasticity abstract a gradient dependent plasticity model is analysed in which the yield function not only depends on the stress tensor and an invariant plastic strain measure but also on the spatial derivatives of the latter quantity a minimum variational principle is obtained by analysing the structure discretized into sufficiently small plastic elements and thus the uniqueness of fe solutions can be ensured for softening plasticity the positive definiteness of the variational functional is determined only by the sign of the diffuse coefficient associated with the highest order derivative of the plastic multiplier in the yield function in order to adopt mix hybrid elements several general variational principles are obtained by relaxing the associated constraints on the displacement strain stress and or plastic multiplier gradient radiation fields

a finite element formulation to solve a non local constitutive model with stresses and strains due to slip gradients solving constitutive models that incorporate the effects of plasticity and slip gradients is often complicated by the non local nature of the models this work presents a finite element solution to a crystal plasticity constitutive model that includes kinematic and stress effects due to slip gradients the foundation of the model is a three term multiplicative decomposition of the deformation gradient that results in a second order differential equation in terms of the stress that drives slip converting the equation into a weak form results in an integral equation that includes first order derivatives of the stress as well as boundary conditions for the stress and gradients of slip rate for each slip system using this weak form an incremental finite element method is developed to solve the constitutive model within a finite element solution to the equilibrium equation results for the compression of a two dimensional plate show the effects of including slip gradient effects in the constitutive model and indicate the tendency for localization of the slip and dislocation density into narrow bands separating regions of nearly constant dislocation density and long range strain

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 generalized hill s lemma and micromechanically based macroscopic constitutive model for heterogeneous granular materials abstract based on the hill s lemma for classical cauchy continuum a generalized hill s lemma for micro macro homogenization modeling of gradient enhanced cosserat continuum is presented in the frame of the average field theory in this context not only the strain and stress tensors defined in classical cosserat continuum but also their gradients are attributed to assigned micro structural representative volume element rve that leads to a higher order macroscopic cosserat continuum modeling and enables to incorporate the micro structural size effects the enhanced hill mandel condition for gradient enhanced cosserat continuum is extracted as a corollary of the presented generalized hill s lemma the derived admissible boundary conditions for the modeling are deduced to direct the proper presentation of boundary conditions to be prescribed on the rve in order to ensure the satisfaction of the hill mandel energy condition with the link between the discrete particle assembly and its effective cosserat continuum in an individual rve the boundary conditions prescribed on the rve modeled as cosserat continuum are transformed into those prescribed to the peripheral particles of the rve modeled as the discrete particle assembly the micromechanically based macroscopic constitutive model and corresponding rate forms of the macroscopic stress strain relations taking into account the local microstructure and its evolution are formulated with neither need of specifying the macroscopic constitutive relation nor need of providing macroscopic material parameters

an algorithm and a method to search bifurcation points in non linear problems this paper is devoted to a numerical method able to help the determination of the bifurcation threshold in non linear time independent continuum mechanic problems first some theoretical results about uniqueness are recalled in the framework of the large strain assumption the differences between the classical finite step problem and the rate problem are presented an iterative algorithm able to solve the rate problem is given using different initializations it is seen in some numerical experiments that it is possible with this algorithm to get different solutions when the underlying mathematical problem solved does not enjoy a uniqueness property the constitutive equations used have been chosen to be simple enough to deduce some theoretical knowledge about the corresponding uniqueness problems finally a method is given which is able in some case to give an upper bound of the bifurcation threshold copyright 2001 john wiley sons ltd

on uniqueness of the dynamic finite step problem in gradient dependent softening plasticity the dynamic evolution of an elastoplastic softening solid is considered a material model including in the yield function the laplacian of the plastic multiplier is used to regularize the problem the dynamic finite step problem is formulated according to a generalized mid point integration scheme space discretization is carried out by a mixed finite element technique based on generalized variables a sufficient uniqueness condition of the finite step solution is proved for a one dimensional problem also a necessary and sufficient condition is presented a simple numerical test shows the regularizing properties mesh independence of the proposed model and the positive influence of the gradient term also on the time step amplitude ensuring uniqueness of solution

post necking behaviour modelled by a gradient dependent plasticity theory abstract the delay of the onset of localization and the post necking behaviour for stretched thin sheets are determined by three dimensional effects thus a 2 d finite element analysis based on a local plasticity theory will give a physically unrealistic mesh dependent solution this in spite of the fact that the stress state is essentially two dimensional by incorporating a length scale with relation to the thickness of the sheet it is demonstrated how a 2 d finite element analysis based on a gradient dependent plasticity theory can give a good approximation of the post necking behaviour this is illustrated by numerical comparison of results from a full 3 d finite element analysis with results from a 2 d finite element model based on a finite strain version of a gradient dependent j 2 flow theory some numerical problems in the modeling will be discussed briefly

theoretical and computational aspects of a thermodynamically consistent framework for geometrically linear gradient damage this paper presents the theory and the numerics of an isotropic gradient damage formulation within a thermodynamical background the main motivation is provided by localization computations whereby classical local continuum formulations fail to produce physically meaningful and numerically converging results we propose a formulation in terms of the helmholtz free energy incorporating the gradient of the damage field a dissipation potential and the postulate of maximum dissipation as a result the driving force conjugated to damage evolution is identified as the quasi nonlocal energy release rate which essentially incorporates the divergence of a vectorial damage flux besides the strictly local energy release rate on the numerical side besides balance of linear momentum the algorithmic consistency condition must be solved in weak form thereby the crucial issue is the selection of active constraints which is solved by an active set search algorithm borrowed from convex nonlinear programming in the examples we compare the behavior in local damage with the performance of the gradient formulation

large strain finite element analysis of a local second gradient model application to localization summary a large strainnite element formulation based on a local second gradient plasticity model is presented the corresponding constitutive equations were developed as a direct extension of microstructured siam j appl math 1973 25 3 556 575 arch rational mech anal 1964 16 51 78 or micromorphic in mechanics of generalized continua iutam symposium kroner ed springer berlin 1968 18 35 j math mech 1966 15 6 909 924 continua in which a mathematical constraint between the micro kinematics description and the usual macrodeformation gradienteld has been introduced this constraint is enforced in a weak sense by the use of lagrange multipliers in order to avoid diculties with the c 1 continuity for thenite element method correspondingnite elements are then constructed involving the lagrange multiplierseld a geometrically non linear 2 dnite element code is developed within a framework of an incremental method for every step a full newton raphson involving a numerical consistent tangent stiness operator for the complete model i e the second gradient terms as well as the classical ones is done and some numerical tests allow to validate the method and to discuss the inuence of the geometrical non linearity copyright 2002 john wiley sons ltd

a refined nonconforming quadrilateral element for couple stress strain gradient elasticity c0 1 patch test int j numer meth engng 2004 61 433 454 proposed by soh and chen is a reliable method to ensure convergence of nonconforming finite element for the couple stress strain gradient elasticity the c0 1 patch test function is a complete quadratic polynomial that satisfies the equilibrium equations to pass the c0 1 patch test the element displacement functions used to calculate strains must satisfy c0 continuity or weak c0 continuity and quadratic completeness in this paper a 24 dof degrees of freedom quadrilateral element cq12 rdkq for the couple stress strain gradient elasticity is developed by combining the refined thin plate element rdkq and the nonconforming element cq12 the element rdkq which satisfies weak c1 continuity is used to calculate strain gradients whereas strains are computed by the element cq12 which is established based on an extended variational functional and satisfies weak c0 continuity and quadratic completeness numerical examples show that the element cq12 rdkq passes the c0 1 patch test and it is also more efficient than the existing available triangular and quadrilateral elements in stress concentration problems with size effects copyright 2010 john wiley sons ltd

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

on the temperature dependence of the mean number of clusters the random variable number of cluster defined on the configurations of a ferromagnetic ising model at zero field and inverse temperature on a graphg is considered the gibbs average at 0 is proved to be greater than the one at 0 if the degree ofg is not greater than 3

an inequality for the gibbs mean number of clusters the number of percolation clusters for configurations of the ising model at zero external field and ferromagnetic first neighbors interaction on a general finite graph is considered the mean number of clusters with respect to the gibbs measure at any inverse temperature is proved to be smaller or equal than the one at 0

unified analysis of enriched finite elements for modeling cohesive cracks abstract aiming to provide a justified theoretical ground based on which different enriched finite elements can be systematically compared we developed in this work a unified framework for modeling cohesive cracks using the variational multiscale method the kinematics i e coarse and fine scale displacement and strain fields and statics i e coarse and fine scale equilibrium equations are thoroughly investigated in both continuum and discrete settings with respect to the fine scale kinematics and statics adopted in the embedded finite element method efem and the extended finite element method xfem we mainly discuss four groups of enriched finite elements with non uniform discontinuity modes i e the kinematically optimal symmetric efem and xfem as well as the kinematically and statically optimal non symmetric efem and xfem in all these methods the enrichment parameters can be regarded either as element wise local or continuous global variables the enriched finite elements with material element node enrichments are then exemplified in an increasing order of their respective capabilities in representing the fine scale kinematics and the interrelations between them are also discussed owing to this unified framework we are able to clarify some existing points of view related to efem and xfem it is found that if the same fine scale statics is used and the same local global property is assumed for the involved enrichment parameters xfem can be regarded as a kinematically enhanced efem since it accounts for a more general fine scale kinematics than that i e relative rigid body motions and self stretching considered in efem finally several simple but sufficiently representative numerical examples are presented to show the significance in appropriately reflecting the fine scale kinematics and statics in an enriched finite element for modeling cohesive cracks

a micro macro homogenization approach for discrete particle assembly cosserat continuum modeling of granular materials based on hill s lemma for classical cauchy continuum a version of hill s lemma for micro macro homogenization modeling of cosserat continuum is derived the derived admissible boundary conditions for the modeling are used to direct the proper prescription of boundary conditions on the representative volume element rve in order to ensure the satisfaction of the hill mandel energy condition a micro macro computational homogenization procedure for modeling the mechanical response of granular material modeled as heterogeneous cosserat continuum is presented taking into account both physical and geometrical non linear evolution of microstructures of the material with the link between the discrete particle assembly and its cosserat continuum equivalent in an individual rve the boundary conditions prescribed on the rve modeled as cosserat continuum are transformed into those prescribed to the peripheral particles of the rve modeled as the discrete particle assembly the average stresses and strains and their rates over the rve defined for the cosserat continuum equivalent are then determined by the physical and geometrical quantities of the discrete particle assembly the consistent macroscopic modulus tensor and the macroscopic constitutive relation defined at the integration point are formulated in terms of the averaged behavior of the associated microstructures since the proposed micro macro computational homogenization procedure is used within the finite element framework there is no need to specify the macroscopic constitutive relation at the macroscopic integration points

arbitrary lagrangian eulerian finite element analysis of strain localization in transient problems non local models guaranty that finite element computations on strain softening materials remain sound up to failure from a theoretical and computational viewpoint the non locality prevents strain localization with zero global dissipation of energy and consequently finite element calculations converge upon mesh refinements to non zero width localization zones one of the major drawbacks of these models is that the element size needed in order to capture the localization zone must be smaller than the internal length hence the total number of degrees of freedom becomes rapidly prohibitive for most engineering applications and there is an obvious need for mesh adaptivity this paper deals with the application of the arbitrary lagrangian eulerian ale formulation well known in hydrodynamics and fluid structure interaction problems to transient strain localization in a non local damageable material it is shown that the ale formulation which is employed in large boundary motion problems can also be well suited for non linear transient analysis of softening materials where localization bands appear the remeshing strategy is based on the equidistribution of an indicator that quantifies the interelement jump of a selected state variable two well known one dimensional examples illustrate the capabilities of this technique the first one deals with localization due to a propagating wave in a bar and the second one studies the wave propagation in a hollow sphere

a variational principle for gradient plasticity abstract we elaborate on a generalized plasticity model which belongs to the class of gradient models suggested earlier by aifantis and co workers the generalization of the conventional theory of plasticity has been accomplished by the inclusion of higher order spatial gradients of the equivalent plastic strain in the yield condition first it is shown how these gradients affect the critical condition for the onset of localization and allow for a wavelength selection analysis leading to estimates for the width and or spacing of shear bands due to the presence of higher order gradients additional boundary conditions for the equivalent plastic strain are required this question and also the associated problem of the formulation and solution of general boundary value problems were left open in the previous work we demonstrate here that upon assuming a certain type of additional boundary conditions the structural symmetries of the gradient dependent constitutive model are such that there exists a variational principle for the displacement rates and the rate of the equivalent plastic strain the variational principle can serve as a basis for the numerical solution of boundary value problems in the sense of the finite element method explicit expressions for the tangent stillness matrix and the generalized nodal point forces are given

modeling strong discontinuities at finite strains a novel numerical implementation abstract in this paper a finite element formulation suitable for the modeling of locally embedded strong discontinuities at finite strains is presented following the enhanced assumed strain concept eas the proposed numerical model is based on an additive decomposition of the displacement gradient into a conforming and an enhanced part the enhanced part is associated with the final failure kinematics of solids which are approximated by means of a discontinuous displacement field strong discontinuities referring to the displacement jump no special assumption such as purely mode i or mode ii failure is made the same holds for the class of interface laws considered which govern the evolution of the displacement discontinuity in terms of the traction vector acting at the surface of strong discontinuities consequently the suggested numerical framework can be applied to a broad range of different interface laws including damage based models in contrast to previous works the presented finite element formulation does not require the static condensation technique to be employed more precisely instead of computing the conforming part of deformation and the displacement jump simultaneously from the weak form of equilibrium and the weak form of traction continuity the different parts of the local deformation are decomposed according to a predictor corrector algorithm the proposed predictor and the corrector step are formally identical to that of classical computational plasticity models hence subroutines designed for standard models continuous deformation can be applied with only minor modifications necessary the applicability as well as the performance of the resulting finite element formulation are demonstrated by means of a fully three dimensional analysis of shear band formation in a bar made of a ductile material

possibilities of finite calculus in computational mechanics abstract the expression finite calculus refers to the derivation of the governing differential equations in mechanics by invoking balance of fluxes forces etc in a domain of finite size the governing equations resulting from this approach are different from those of infinitessimal calculus theory and they incorporate new terms depending on the dimensions of the balance domain the new modified equations allow to derive naturally stabilized numerical schemes using finite element finite difference finite volume or meshless methods the paper briefly discusses the possibilities of the modified governing equations derived via the finite calculus technique for the numerical solution of convection diffusion problems incompressible flow and incompressible solid mechanic problems and strain localization problems

numerical approximation of incremental infinitesimal gradient plasticity we investigate a representative model of continuum infinitesimal gradient plasticity the formulation is an extension of classical rate independent infinitesimal plasticity based on the additive decomposition of the symmetric strain tensor into elastic and plastic parts it is assumed that dislocation processes contribute to the storage of energy in the material whereby the curl of the plastic distortion appears in the thermodynamic potential and leads to an additional nonlocal backstress tensor the formulation is cast into a numerical framework by a saddle point approximation of the corresponding minimization problem in each incremental loading step this allows one to reformulate the nonlocal dissipation inequality to a point wise flow rule and yields a solution scheme which is a direct extension of the standard approach in classical plasticity our numerical results show the regularizing effects of the additional physically motivated terms copyright 2008 john wiley sons ltd

three dimensional isogeometric solutions to general boundary value problems of toupin s gradient elasticity theory at finite strains abstract we present to the best of our knowledge the first complete three dimensional numerical solutions to a broad range of boundary value problems for a general theory of finite strain gradient elasticity we have chosen for our work toupin s theory toupin 1962 one of the more general formulations of strain gradient elasticity our framework has three crucial ingredients the first is isogeometric analysis hughes et al 2005 which we have adopted for its straightforward and robust representation of c 1 continuity the second is a weak treatment of the higher order dirichlet boundary conditions in the formulation which control the development of strain gradients in the solution the third ingredient is algorithmic automatic differentiation which eliminates the need for linearization by hand of the rather complicated geometric and material nonlinearities in gradient elasticity at finite strains we present a number of numerical solutions to demonstrate that the framework is applicable to arbitrary boundary value problems in three dimensions we discuss length scale effects the role of higher order boundary conditions and perhaps most importantly the relevance of the framework to problems with elastic free energy density functions that are non convex in strain space

shear band localization via local j2 continuum damage mechanics this paper describes a novel formulation for the solution of problems involving shear band localization using a local isotropic j2 continuum damage model and mixed linear simplex triangles and tetrahedra stabilization methods are used to ensure existence and uniqueness of the solution attaining global and local stability of the corresponding discrete finite element formulation consistent residual viscosity is used to enhance robustness and convergence of the formulation implementation and computational aspects are also discussed a simple isotropic local j2 damage constitutive model is considered either with linear or exponential softening the softening modulus is regularized according to the material mode ii fracture energy and the element size numerical examples show that the formulation derived is fully stable and remarkably robust totally free of volumetric locking and spurious oscillations of the pressure as a consequence the results obtained do not suffer from spurious mesh size or mesh bias dependence comparing very favourably with those obtained with the ill posed standard approaches 2004 elsevier b v all rights reserved

an isogeometric analysis approach to gradient dependent plasticity summary gradient dependent plasticity can be used to achieve mesh objective results upon loss of well posedness of the initial boundary value problem due to the introduction of strain softening non associated flow and geometric non linearity a prominent class of gradient plasticity models considers a dependence of the yield strength on the laplacian of the hardening parameter usually an invariant of the plastic strain tensor this inclusion causes the consistency condition to become a partial differential equation in addition to the momentum balance at the internal moving boundary one has to impose appropriate boundary conditions on the hardening parameter or equivalently on the plastic multiplier this internal boundary condition can be enforced without tracking the elastic plastic boundary by requiring c1 continuity with respect to the plastic multiplier in this contribution this continuity has been achieved by using nurbs as shape functions both for the plastic multiplier and for the displacements one advantage of this isogeometric analysis approach is that the displacements can be interpolated one order higher making it consistent with the interpolation of the plastic multiplier this is different from previous approaches which have been exploited the regularising effect of gradient plasticity is shown for one and two dimensional boundary value problems this article is protected by copyright all rights reserved

a discontinuous galerkin formulation for classical and gradient plasticity part 2 algorithms and numerical analysis this work is the second of a two part investigation into the use of discontinuous galerkin methods for obtaining approximate solutions to problems of classical and gradient plasticity part i j k djoko f ebobisse a t mcbride b d reddy a discontinuous galerkin formulation for classical and gradient plasticity part 1 formulation and analysis comput methods appl mech engrg 196 2007 3881 3897 presented the formulation and analysis of such problems this part focusses on algorithmic and computational aspects of the problem in particular it is shown that the predictor corrector algorithms of classical plasticity are readily extended to the case of gradient plasticity and to discontinuous galerkin formulations conditions for convergence of the algorithms are presented for the elastic secant and consistent tangent predictors the form of the consistent tangent modulus is established for the case of gradient plasticity a selection of numerical examples is presented and discussed with a view to illustrating aspects of the approximation scheme and algorithms as well as features of the model of gradient plasticity adopted here

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