stress gradient versus strain gradient constitutive models within elasticity abstract a stress gradient elasticity theory is developed which is based on the eringen method to address nonlocal elasticity by means of differential equations by suitable thermodynamics arguments involving the free enthalpy instead of the free internal energy the restrictions on the related constitutive equations are determined which include the well known eringen stress gradient constitutive equations as well as the associated so far uncertain boundary conditions the proposed theory exhibits complementary characters with respect to the analogous strain gradient elasticity theory the associated boundary value problem is shown to admit a unique solution characterized by a hellinger reissner type variational principle the main differences between the eringen stress gradient model and the concomitant aifantis strain gradient model are pointed out a rigorous formulation of the stress gradient euler bernoulli beam is provided the response of this beam model is discussed as for its sensitivity to the stress gradient effects and compared with the analogous strain gradient beam model

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

on asymptotics in case of linear index 2 differential algebraic equations asymptotic properties of solutions of general linear differential algebraic equations daes and those of their numerical counterparts are discussed new results on the asymptotic stability in the sense of lyapunov as well as on contractive index 2 daes are given the behavior of the backward differentiation formula bdf implicit runge kutta irk and projected implicit runge kutta pirk methods applied to such systems is investigated in particular we clarify the significance of certain subspaces closely related to the geometry of the dae asymptotic properties like a stability and l stability are shown to be preserved if these subspaces are constant moreover algebraically stable irk dae are b stable under this condition the general results are specialized to the case of index 2 hessenberg systems

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

adjoint sensitivity analysis for differential algebraic equations the adjoint dae system and its numerical solution an adjoint sensitivity method is presented for parameter dependent differential algebraic equation systems daes the adjoint system is derived along with conditions for its consistent initialization for daes of index up to two hessenberg for stable linear daes stability of the adjoint system for semi explicit daes or of an augmented adjoint system for fully implicit daes is shown in addition it is shown for these systems that numerical stability is maintained for the adjoint system or for the augmented adjoint system

strain gradient theory in orthogonal curvilinear coordinates in this short note general formulations of the toupin mindlin strain gradient theory in orthogonal curvilinear coordinate systems are derived and are then specified for the cases of cylindrical coordinates and spherical coordinates expressions convenient for practical use are presented for the corresponding equilibrium equations boundary conditions and the physical components for strains and strain gradients in the two coordinate systems the results obtained in this paper are general and complete and can be useful for a wide range of applications such as asymptotic crack tip field analysis cylindrical and spherical cavity expansion problems and the interpretation of micro nano indentation tests and bending twisting tests on small scales

on index 2 linear implicit difference equations this paper deals with an index 2 notion for linear implicit difference equations lides and with the solvability of initial value problems ivps for index 2 lides besides the cocycle property as well as the multiplicative ergodic theorem of oseledets type are also proved

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

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

on a question of quillen let r be a regular local ring and f a regular parameter of r quillen asked whether every projective r f module is free we settle this question when r is a regular local ring of an affine algebra over a field k further if r has infinite residue field we show that projective modules over laurent polynomial extensions of r f are also free

a micropolar formulation of the desai hierarchical model for elastoplastic porous media in this contribution the desai hierarchical model is extended to the case of 3d elasto plastic two phase micropolar continuum an unconditionally stable implicit euler backward algorithm for integration of the constitutive relations is developed and presented in detail the regularizing effect of the introduction of a length parameter to the classical desai model is demonstrated by means of several case studies of single phase and two phase porous media

projected implicit runge kutta methods for differential algebraic equations in this paper a new class of numerical methods projected implicit runge kutta methods is introduced for the solution of index 2 hessenberg systems of initial and boundary value differential algebraic equations daes these types of systems arise in a variety of applications including the modeling of singular optimal control problems and parameter estimation for differential algebraic equations such as multibody systems the new methods appear to be particularly promising for the solution of dae boundary value problems where the need to maintain stability in the differential part of the system often necessitates the use of methods based on symmetric discretizations previously defined symmetric methods have severe limitations when applied to these problems including instability oscillation and loss of accuracy the new methods overcome these difficulties for linear problems we define an essential underlying boundary value ode and prove well conditioning of the differential or state space solutio

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

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

stability of numerical methods for solving linear index 3 daes in this paper the solvability of linear index 3 daes and the stability of the numerical methods are investigated by choosing suitable projectors the solutions of daes are decomposed into their essential components some equivalent systems and consistent initial conditions are given in a similar way the stability of the bdfs and irk methods applied to some daes is discussed

stability radii of differential algebraic equations with structured perturbations abstract this paper deals with a formula for computing stability radii of a differential algebraic equation of the form a x t b x t 0 where a b are constant matrices a computable formula for the complex stability radius is given and a key difference between the ordinary differential equation odes for short and the differential algebraic equation daes for short is pointed out a special case where the real stability radius and the complex one are equal is considered

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

projector based integration of daes with the taylor series method using automatic differentiation automatic or algorithmic differentiation ad opens new possibilities to analyze and solve daes by projector based methods in this paper we present a new approach to compute consistent initial values and integrate daes up to index two considering the nonlinear dae in each time step as a nonlinear system of equations for taylor expansions these systems will be solved by the newton kantorowitsch method whereas the resulting linear systems are decoupled using the splitting techniques related to the tractability index concept this approach provides a description of the inherent ode that allows an application of the classical taylor series method to the integration of initial value problems linear and nonlinear daes with index up to two are examined and solved numerically

a direct finite element implementation of the gradient dependent theory the enhanced non local gradient dependent theories formulate a constitutive framework on the continuum level that is used to bridge the gap between the micromechanical theories and the classical local continuum they are successful in explaining the size effects encountered at the micron scale and in preserving the well posedeness of the i bvp governing the solution of material instability triggering strain localization this is due to the incorporation of an intrinsic material length scale parameter in the constitutive description however the numerical implementation of these theories is not a direct task because of the higher order of the governing equations in this paper a direct computational algorithm for the gradient approach is proposed this algorithm can be implemented in the existing finite element codes without numerous modifications as compared to the current numerical approaches int j solids struct 1988 24 581 597 int j numer meth engng 1992 35 521 539 eng comput 1993 10 99 121 dissertation 1994 int j numer meth engng 1996 39 2477 2505 int j numer meth engng 1996 39 3731 3755 comput meth appl mech eng 1998 163 11 32 comput meth appl mech eng 1998 163 33 53 euro j mech a solids 1999 18 939 962 int j solids struct 2000 37 7481 7499 a predictor corrector scheme is proposed for the solution of the non linear algebraic problem from the fem the expressions of the continuum and consistent tangent matrices are provided the method is validated by conducting various numerical tests as a result pathological mesh dependence as obtained in finite element computations with conventional continuum models is no longer encountered copyright 2005 john wiley sons ltd

a micromechanics based approach for the derivation of constitutive elastic coefficients of strain gradient media a micromechanics based approach for the derivation of the effective properties of periodic linear elastic composites which exhibit strain gradient effects at the macroscopic level is presented at the local scale all phases of the composite obey the classic equations of three dimensional elasticity but since the assumption of strict separation of scales is not verified the macroscopic behavior is described by the equations of strain gradient elasticity the methodology uses the series expansions at the local scale for which higher order terms which are generally neglected in standard homogenization framework are kept in order to take into account the microstructural effects an energy based micro macro transition is then proposed for upscaling and constitutes in fact a generalization of the hill mandel lemma to the case of higher order homogenization problems the constitutive relations and the definitions for higher order elasticity tensors are retrieved by means of the state law associated to the derived macroscopic potential as an illustration purpose we derive the closed form expressions for the components of the gradient elasticity tensors in the particular case of a stratified periodic composite for handling the problems with an arbitrary microstructure a fft based computational iterative scheme is proposed in the last part of the paper its efficiency is shown in the particular case of composites reinforced by long fibers

order results for implicit runge kutta methods applied to differential algebraic systems in this paper we study the order stability and convergence properties of implicit runge kutta methods applied to a relatively simple class of nonlinear differential algebraic systems these methods often do not attain the same order of accuracy for differential algebraic systems as they do for purely differential systems we derive a set of order conditions which the method coefficients should satisfy in addition to the usual order conditions to ensure a given order of accuracy and we present results on the stability and convergence properties of these methods

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

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

a note on the higher order strain and stress tensors within deformation gradient elasticity theories physical interpretations and comparisons abstract higher order strain and stress tensors encompassed within gradient elasticity theories are discussed with a particular concern to the physical meaning of double and triple stresses a single rule is shown to hold for the physical interpretation of the indices of a higher order stress tensor both within distortion gradient and strain gradient theories whereas the analogous mindlin s rule holds only within distortion gradient theories double and triple stresses are discussed separately with the aid of simple illustrative examples a corrigendum to a previous paper by the author ijss 50 2013 3749 3765 is also presented

stability of computational methods for constrained dynamics systems many methods have been proposed for numerically integrating the differential algebraic systems arising from the euler lagrange equations for constrained motion these are based on various problem formulations and discretizations we offer a critical evaluation of these methods from the standpoint of stability nconsidering a linear model we first give conditions under which the differential algebraic problem is well conditioned this involves the concept of an essential underlying ode we review a variety of reformulations which have been proposed in the literature and show that most of them preserve the well conditioning of the original problem then we consider stiff and nonstiff discretizations of such reformulated models in some cases the same implicit discretization may behave in a very different way when applied to different problem formulations acting as a stiff integrator on some formulations and as a nonstiff integrator on others we present the approach of projected invariants as a method for yielding problem reformulations which are desirable in this sense

time and space adaptive methods applied to localization phenomena in empty and saturated micropolar and standard porous materials localization phenomena as e g shear bands occur as a result of local concentrations of plastic strains in small bands of finite width the reason for this behaviour lies in the basic properties of elasto plastic frictional materials where variations of the lode angle together with the non associated plastic dilatation result in local softening effects proceeding from non viscous empty porous materials it is well known that the computation of shear bands reveals an ill posed problem to overcome this behaviour two different regularization strategies are considered which are 1 the inclusion of independent rotational degrees of freedom micropolar formulation and 2 the inclusion of viscosity effects standard formulation by taking into consideration viscoplastic properties of the solid matrix or a viscous pore fluid respectively the numerical treatment furthermore proceeds from time and space adaptive strategies to refine and to coarsen both the time step and the mesh size considering quasi static problems the space discretization yields a dae system of index 1 in the time domain which can be successfully treated by sdirk methods thus allowing for an efficient estimation of the time error and as a consequence for an adaptive time step control in the space domain the present investigation proceeds from an error indicator of zienkiewicz zhu type where smoothened values of the l2 norms of characteristic quantities are compared to the respective discrete values the efficiency of this procedure is demonstrated by the computation of the biaxial experiment on an empty and a liquid saturated porous soil material and of the slope failure problem of a fluid saturated soil copyright 2001 john wiley sons ltd

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

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

non local yield limit degradation presented is a new type of a non local continuum model which avoids problems of convergence at mesh refinement and spurious mesh sensitivity in a softening continuum characterized by degradation of the yield limit the key idea which has recently been proposed in a general context and has already been applied to softening damage due to stiffness degradation is to apply the non local concept only to those parameters which cause the degradation while keeping the definition of the strains local compared to the previously advanced fully non local continuum formulation the new approach has the advantage that the stresses are subjected to the standard differential equations of equilibrium and standard boundary or interface conditions the new formulation exhibits no zero energy periodic modes imbrication of finite elements is unnecessary and finite elements with standard continuity requirements are sufficient two dimensional finite element solutions with up to 3248 degrees of freedom are presented to document convergence and efficacy the formulation is applied to tunnel excavation in a soil stabilized by cement grouting with the objective of preventing cave in burst of the tunnel sides due to compression softening

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

microdifferential systems and the codimension three conjecture in this paper we give a proof of a fundamental conjecture the codimension three conjecture for microdierential holonomic systems the conjecture states that any regular holonomic module extends uniquely beyond an analytic subset that is at least of codimension three in its support our result can also be interpreted from a topological point of view as a statement about microlocal perverse sheaves however our proof is entirely in the context of microdierential holonomic systems

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

infinitesimal elastic plastic cosserat micropolar theory modelling and global existence in the rate independent case in this article we investigate the regularizing properties of generalized continua of cosserat micropolar type in the elasto plastic case we propose an extension of classical infinitesimal elasto plasticity to include consistently non dissipative micropolar effects it is shown that the new model is thermodynamically admissible and allows for a unique global in time solution of the corresponding rate independent initial boundary value problem the method of choice is the yosida approximation and a passage to the limit

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

variational formulations convergence and stability properties in nonlocal elastoplasticity a thermodynamically consistent formulation of nonlocal plasticity in the framework of the internal variable theories of inelastic behaviors of associative type is presented a family of mixed variational formulations with different combinations of state variables is provided starting from the finite step nonlocal elastoplastic structural problem it is shown that a suitable minimum principles provides a rational basis to exploit the iterative elastic predictor plastic corrector algorithm in terms of the dissipation functional a sufficient condition is proved for the convergence of the iterative elastic predictor plastic corrector algorithm based on a suitable choice of the elastic operator in the prediction phase and a necessary and sufficient condition for the existence of a unique solution if any of the nonlocal problem at hand is then provided the nonlinear stability analysis of the nonlocal problem is carried out following the concept of nonexpansivity proposed in local plasticity

solving more general index 2 differential algebraic equations abstract the paper deals with a class of quasilinear index 2 differential algebraic equations which covers both linear variable coefficient systems as well as hessenberg form equations supposing low smoothness only the solvability of initial value problems is stated via classical analytical techniques for that class of differential algebraic equations backward differentiation formulas and runge kutta methods as well as projected versions are discussed with respect to feasibility in stability convergence and asymptotical behaviour

global existence results for viscoplasticity at finite strain we study a model for rate dependent gradient plasticity at finite strain based on the multiplicative decomposition of the strain tensor and investigate the existence of global in time solutions to the related pde system we reveal its underlying structure as a generalized gradient system where the driving energy functional is highly nonconvex and features the geometric nonlinearities related to finite strain elasticity as well as the multiplicative decomposition of finite strain plasticity moreover the dissipation potential depends on the left invariant plastic rate and thus depends on the plastic state variable the existence theory is developed for a class of abstract nonsmooth and nonconvex gradient systems for which we introduce suitable notions of solutions namely energy dissipation balance and energy dissipation inequality solutions hence we resort to the toolbox of the direct method of the calculus of variations to check that the specific energy and dissipation functionals for our viscoplastic models comply with the conditions of the general theory

analyzing the stability behaviour of solutions and their approximations in case of index 2 differential algebraic systems when integrating regular ordinary differential equations numerically one tries to match carefully the dynamics of the numerical algorithm with the dynamical behaviour of the true solution the present paper deals with linear index 2 differential algebraic systems it is shown how knowledge pertaining to numerical regular ordinary differential equations applies provided a certain subspace which is closely related to the tangent space of the constraint manifold remains invariant

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

energy relaxation of non convex incremental stress potentials in a strain softening elastic plastic bar abstract we propose a fundamentally new concept to the treatment of material instabilities and localization phenomena based on energy minimization principles in a strain softening elastic plastic bar the basis is a recently developed incremental variational formulation of the local constitutive response for generalized standard media it provides a quasi hyperelastic stress potential that is obtained from a local minimization of the incremental energy density with respect to the internal variables the existence of this variational formulation induces the definition of the material stability of inelastic solids based on convexity properties in analogy to treatments in elasticity furthermore localization phenomena are understood as micro structure development associated with a non convex incremental stress potential in analogy to phase decomposition problems in elasticity for the one dimensional bar considered the two phase micro structure can analytically be resolved by the construction of a sequentially weakly lower semicontinuous energy functional that envelops the not well posed original problem this relaxation procedure requires the solution of a local energy minimization problem with two variables which define the one dimensional micro structure developing the volume fraction and the intensity of the micro bifurcation the relaxation analysis yields a well posed boundary value problem for an objective post critical localization analysis the performance of the proposed method is demonstrated for different discretizations of the elastic plastic bar which document on the mesh independence of the results

meshfree analysis of softening elastoplastic solids using variational multiscale method a meshfree multiscale method is presented for efficient analysis of elastoplastic solids in the analysis of softening elastoplastic solids standard finite element methods or meshfree methods typically yield mesh dependent results the reason for this well known effect is the loss of ellipticity of the boundary value problem in this work the scale decomposition is carried out based on a variational form of the problem a coarse scale is designed to represent global behavior and a fine scale to represent local behavior a fine scale region is detected from the local failure analysis of an acoustic tensor to indicate a region where deformation changes abruptly each scale variable is approximated using a meshfree method meshfree approximation is well suited for adaptivity as a method of increasing the resolution a partition of unity based extrinsic enrichment is used in particular fine scale approximations are designed to appropriately represent local behavior by using a localization angle moreover the regularization effect through the convexification of non convex potential is embedded to represent fine scale behavior each scale problem is solved iteratively the proposed method is applied to shear band problems in the results of analysis about pure shear and compression problems straight shear bands can be captured and mesh insensitive results are obtained curved shear bands can also be captured without mesh dependency in the analysis of indentation problem

a formulation for the micromorphic continuum at finite inelastic strains the paper presents a generalized theory of deformation which can capture scale effects also in a homogenously deforming body scale effects are relevant when the dimensions of the specimen or structure themselves are in the micron and submicron scale but also when it comes to high strain concentrations as in the case of localised shear bands or at crack tips etc in this context so called generalized continuum formulations have been proven to provide remedy as they allow for the incorporation of internal length scale parameters which reflect the micro structural influence on the macroscopic material response here we want to adopt a generalized continuum framework which is based on the mathematical description of a combined macro and micro space sansour 1998b the approach introduces additional degrees of freedom which constitute a so called micromorphic deformation first the treatment presented is general in nature but will be specified for the sake of an example and the number of extra degrees of freedom will be reduced to four based on the generalized deformation description new strain and stress measures are defined which lead to the formulation of a corresponding generalized variational principle the theory is completed by dirichlet boundary conditions for the displacement field of great advantage is the fact that the constitutive law is defined in the generalized space but can be classical otherwise this limits the number of the extra material parameters necessary to those needed for the specification of the micro space in the example presented to only one an example of scale effects in a homogenously deforming specimen and a further example of shearband formation are presented where the constitutive law is a classical viscoplastic one

gradient damage models toward full scale computations abstract the description of localisation and structural failure through local damage constitutive laws leads to ill posed problems to retrieve a well posed problem the material state is described through the gradients of damage in addition to the strain and the damage under some assumptions it is shown that these gradient laws can be cast into a variational formulation its numerical treatment is reduced to the design of a mixed finite element so that algorithms such as newton s method and path following techniques are still available possible convergence difficulties related to the reduction of the damage band width and the appearance of micro instabilities are removed thanks to the design of a new class of brittle laws finally numerical applications show the convergence with mesh refinement the compatibility with coarser approaches such as griffith s theory and cohesive crack models the robustness in case of unstable multi cracking the applicability to full scale 3d situations and the physical relevance compared to experimental data

limit theorems for gradient dependent elastoplastic geomaterials in this paper the limit theorems proposed by drucker and co workers are reformulated to address a class of gradientdependent elastoplastic geomaterials the gradient effects are accounted for by incorporating strain gradients and their conjugate higher order stresses into the constitutive descriptions gradient dependent equilibrium equations and higher order boundary conditions are defined for both statically admissible and kinematically admissible states and the associated lower bound and upper bound theories are recast for the gradient enhanced limit theorems a generalised drucker prager yield criterion that includes the gradient influence on the deviatoric stress is proposed and then employed to investigate the collapse loads for soil layers under conditions of generalised plane strain simple shear and uniaxial compression the corresponding lower and upper bounds are found for these problems it is demonstrated that the predicted collapse bounds are generally dependent on both the conventional and gradient properties with normalised length scale s being present in the results this feature enables us to give physically reasonable interpretations for size effects and shear banding during material collapse comparisons are also made between the gradient dependent bounds and those obtained through conventional plasticity theories influences of model parameters and sample dimensions on the predictions are also discussed it is shown that the proposed gradient dependent limit theorems can be used to provide physically meaningful predictions for general geotechnical applications 2006 elsevier ltd all rights reserved

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

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

least squares collocation for linear higher index differentialalgebraic equations differentialalgebraic equations with higher index give rise to essentially ill posed problems therefore their numerical approximation requires special care in the present paper we state the notion of ill posedness for linear differentialalgebraic equations more precisely based on this property we construct a regularization procedure using a least squares collocation approach by discretizing the pre image space numerical experiments show that the resulting method has excellent convergence properties and is not much more computationally expensive than standard collocation methods used in the numerical solution of ordinary differential equations or index 1 differentialalgebraic equations convergence is shown for a limited class of linear higher index differentialalgebraic equations

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

disturbed state constitutive modeling based on stress strain and nondestructive behavior the proposed disturbed state concept dsc is based on the idea that a deforming material element can be treated as a mixture of two constituent parts in the relative intact ri and fully adjusted fa states referred to as reference states during external loading the material experiences internal changes in its microstructure due to a process of self adjustment and as a consequence the initial ri state transforms continuously to the fa state the self adjustment process which may involve relative motions of material particles that can lead to microcracking and damage or strengthening can cause disturbances in the observed response with respect to the responses under the two reference states then the observed response is expressed in terms of the responses for the ri and fa states that are determined from laboratory tests on material specimens the dsc permits development of unified constitutive models that include hierarchically other previous continuum and damage models as special cases various aspects of the dsc are verified here with respect to laboratory behavior of two materials a cemented sand and a ceramic composite some of the unique features of this study are that 1 the constitutive behavior and parameters can be obtained from the stress strain volume change behavior and from the measurements of ultrasonic p wave velocities 2 correlations between mechanical and ultrasonic response can be established 3 the concept can provide a description of the crack density and 4 it can be simplified for predicting the remaining life of materials through definition of constitutive models and evaluation of design moduli affected by mechanical and environmental loading

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