mixed variational principles and robust finite element implementations of gradient plasticity at small strains summary this work outlines a theoretical and computational framework of gradient plasticity based on a rigorous exploitation of mixed variational principles in contrast to classical local approaches to plasticity based on locally evolving internal variables order parameter fields are taken into account governed by additional balance type pdes including micro structural boundary conditions this incorporates non local plastic effects based on length scales which reflect properties of the material micro structure we develop a unified variational framework based on mixed saddle point principles for the evolution problem of gradient plasticity which is outlined for the simple model problem of von mises plasticity with gradient extended hardening softening response the mixed variational structure includes the hardening softening variable itself as well as its dual driving force the numerical implementation exploits the underlying variational structure yielding a canonical symmetric structure of the monolithic problem it results in a novel finite element fe design of the coupled problem incorporating a long range hardening softening parameter and its dual driving force this allows a straightforward local definition of plastic loading unloading driven by the long range fields providing very robust fe implementations of gradient plasticity this includes a rational method for the definition of elastic plastic boundaries in gradient plasticity along with a post processor that defines the plastic variables in the elastic range we discuss alternative mixed fe designs of the coupled problem including a local global solution strategy of short range and long range fields this includes several new aspects such as extended q1p0 type and mini type finite elements for gradient plasticity all methods are derived in a rigorous format from variational principles numerical benchmarks address advantages and disadvantages of alternative fe designs and provide a guide for the evaluation of simple and robust schemes for variational gradient plasticity copyright 2013 john wiley sons ltd

some novel developments in finite element procedures for gradient dependent plasticity summary improved algorithms are proposed for a gradient plasticity theory in which the laplacian of an invariant plastic strain measure enters the yield function particular attention is given to the type of finite elements that can be used within the format of gradient dependent plasticity assuming a weak satisfaction of the yield function mixed finite elements are developed in which the invariant plastic strain measure and the displacements are discretized two families of finite elements are developed one in which the invariant plastic strain measure is interpolated using c continuous polynomials and one in which penalty enhanced c continuous interpolants are used the performance of both families of finite elements is assessed numerically in one dimensional and two dimensional boundary value problems the regularizing effect of the used gradient enhancement in computations of elastoplastic solids is demonstrated both for mesh refinement and for the directional bias of the grid lines

bounds on the effective response for gradient crystal inelasticity based on homogenization and virtual testing this paper presents the application of variationally consistent selective homogenization applied to a polycrystal with a subscale model of gradient enhanced crystal inelasticity although the full gradient problem is solved on statistical volume elements sves the resulting macroscale problem has the formal character of a standard local continuum a semi dual format of gradient inelasticity is exploited whereby the unknown global variables are the displacements and the energetic micro stresses on each slip system the corresponding time discrete variational formulation of the sve problem defines a saddle point of an associated incremental potential focus is placed on the computation of statistical bounds on the effective energy based on virtual testing on sves and an argument of ergodicity as it turns out suitable combinations of dirichlet and neumann conditions pertinent to the standard equilibrium and the micro force balance respectively will have to be imposed whereas arguments leading to the upper bound are quite straightforward those leading to the lower bound are significantly more involved hence a viable approximation of the lower bound is computed in this paper numerical evaluations of the effective strain energy confirm the theoretical predictions furthermore heuristic arguments for the resulting macroscale stress strain relations are numerically confirmed

a gradient enhanced large deformation continuum damage model for fibre reinforced materials a non local gradient based damage formulation within a geometrically non linear setting is presented the hyperelastic constitutive response at local material point level is governed by a strain energy which is additively composed of an isotropic matrix and of an anisotropic fibre reinforced material respectively the inelastic constitutive response is governed by a scalar 1 d type damage formulation where only the anisotropic elastic part is assumed to be affected by the damage following the concept in dimitrijevic and hackl 28 the local free energy function is enhanced by a gradient term this term essentially contains the gradient of the non local damage variable which itself is introduced as an additional independent variable in order to guarantee the equivalence between the local and non local damage variable a penalisation term is incorporated within the free energy function based on the principle of minimum total potential energy a coupled system of euler lagrange equations i e the balance of linear momentum and the balance of the non local damage field is obtained and solved in weak form the resulting coupled highly non linear system of equations is symmetric and can conveniently be solved by a standard incremental iterative newton raphson type solution scheme several three dimensional displacement and force driven boundary value problems partially motivated by biomechanical application highlight the mesh objective characteristics and constitutive properties of the model and illustratively underline the capabilities of the formulation proposed

a discontinuous galerkin formulation for classical and gradient plasticity part 1 formulation and analysis a discontinuous galerkin formulation is developed and analyzed for the cases of classical and gradient plasticity the model of gradient plasticity is based on the von mises yield function in which dependence is on the isotropic hardening parameter and its laplacian the problem takes the form of a variational inequality of the second kind the discontinuous galerkin formulation is shown to be consistent and convergent error estimates are obtained for the cases of semi and fully discrete formulations these mimic the error estimates obtained for classical plasticity with the conventional galerkin formulation

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

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

thermodynamics based gradient plasticity theories with an application to interface models in the framework of small deformations the so called residual based gradient plasticity theory is reconsidered and improved using the notion of moving geometrically necessary dislocations gnds suitable micromechanics interpretations are heuristically given for the higher order boundary conditions and the long distance particle interactions also a comparison is made between this theory and the analogous virtual work principle vwp based one whereby their respective conceptual and methodological features are pointed out the conditions under which the two theories lead to a same constitutive model are investigated showing that correspondingly a certain indeterminacy exhibited by the vwp based theory disappears a phenomenological interface model is used to better point out differences and analogies of the two theories

admittance of an isolated waveguide fed slot radiating between baffles using a spectrum of two dimensional solutions an efficient method of computing resonant length and admittance characteristics of an isolated broad wall shunt slot radiating between baffles of finite height is presented the outer three dimensional 3d field problem associated with this geometry is reduced to a two dimensional 2d one via a fourier transformation with respect to the longitudinal z direction for each value of the longitudinal wave number k sub z an integral equation is solved for the e field in the mouth of the plates using the method of moments this procedure is repeated for several discrete values of k sub z to obtain a spectrum of 2d solutions which are then inverse transformed to construct the 3d solution in the spatial domain for the exterior baffle region and the half space the slot aperture field is determined by the conventional moment method solution to the integral equation that enforces the continuity of the h field across the slot scattering properties of the slot are then deduced numerical results for the resonant length and resonant conductance are presented computer results are found to be in good agreement with experimentally measured data

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

analysis of a dielectric covered waveguide slot antenna using the spectrum of two dimensional solutions this communication presents an effective and accurate method of computing admittance and resonant characteristics of a single longitudinal waveguide slot radiating into a dielectric layer the external magnetic field is computed using the spectrum of two dimensional solutions s2ds which reduces a three dimensional 3d problem to a 2d problem s2ds method avoids involving double singular integrals while it only deals with single integral with simple point singularity which needs much less computational effort the spectrum of the 2d solution is then inverse fourier transformed to the spatial 3d one the slot field is determined by galerkin s method of moments mom then the scattering coefficients and thus the resonant length and conductance are extracted computed results are in good agreement with other methods and measurement data

green s function for an infinite slot printed between two homogeneous dielectrics i magnetic currents this sequence of papers presents an analytical closed form for the green s function of an infinite slot printed between two different homogeneous dielectrics this first part is devoted to the derivation of the slot magnetic current and to the discussion of the relevant physical implications the fourier spectrum of the magnetic current is derived by solving in analytical form under small width approximation the integral equation ie representing the continuity of the magnetic field trough the slot axis the accuracy of the result is validated trough a fine meshing method of moments from the consequent spectral expression a closed form approximation of the leaky wave propagation and attenuation constants is derived an asymptotic expression for the current is also obtained by steepest descent path evaluation of the pertinent spectral integral analytical expressions of the quasi static reactive contribution is given for both elementary dipole and delta gap excitations

localization properties of strain softening gradient plasticity models part i strain gradient theories this paper compares and evaluates strain gradient extensions of the conventional plasticity theory attention is focused on the ability of individual formulations to act as localization limiters i e to regularize the boundary value problem in the presence of softening and to prevent localization of plastic strain increments into a set of zero measure to keep the presentation simple and to highlight the essential properties of the investigated models only the static rate independent response in the small strain range and in the one dimensional setting is considered these restrictions permit an analytical or semi analytical treatment of the problem while the basic characteristics of the solutions remain valid in the general multi dimensional case the onset of localization is characterized as a bifurcation from a uniform state the subsequent evolution of the localized process zone and of the shape of the strain profile is studied numerically it is shown that certain pathologies e g expansion of the plastic region accompanied by stress locking may arise at later stages of localization a similar analysis of models with gradients of internal variables is presented in a companion paper

a numerical energy conserving method for the dnls equation an implicit numerical energy conserving method is developed for the derivative nonlinear schrodinger dnls equation for periodic boundary conditions we find no numerical high frequency modulational instabilities in addition to the modulational instability from a linear analysis around a nonlinear state for the dnls equation if the modulation is small and k0 a22 2 t k0 is the wavenumber and a the amplitude the numerical scheme is used to follow the nonlinear behavior of the dnls modulational instability the numerical code is also tested by the evolution for one soliton initial data these tests show that if the modulation is not small compared to the background wave amplitude new nonlinear numerical instabilities are introduced

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

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

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

implicit gradient elasticity a new formulation of gradient elasticity is derived and implemented pade approximations are used to introduce an implicit relation between non local strains and displacements as a result the finite element interpolation requires only 0 continuous rather than 1 continuous shape functions the underlying energy functional is presented and it is found that the present formulation is of the mixed type whereby the non local strains act as the primary unknowns and the displacements as the auxiliary unknowns the implications on the interpolation are studied finally the influence of the additional length scale parameter on the global response is assessed copyright 2006 john wiley sons ltd

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

mutual admittance between slots in cylinders of arbitrary shape a numerical formulation involving the fourier transform and the combined field integral equation is presented to calculate the mutual admittance between axial slots or circumferential slots in arbitrarily shaped cylinders two cases are considered coupling between two circumferential slots and coupling between two axial slots the slots are assumed to be waveguide fed apertures excited with the te sub 10 waveguide mode with no higher order modes included in the model the combined field integral equation cfie formulation was used in conjunction with the magnetic field integral equation mfie to solve the resulting two dimensional integral equation formulations for the axial slot case are discussed results concerning the discretization and truncation of the spectrum as well as the accuracy of the procedure are presented for both axial and circumferential cases the accuracy of the procedure is demonstrated by comparison with eigenfunction data for circular cylinders

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

closed form green s function representations in cylindrically stratified media for method of moments applications closed form green s function cfgf representations for cylindrically stratified media which can be used as the kernel of an electric field integral equation are developed the developed cfgf representations can safely be used in a method of moments solution procedure as they are valid for almost all possible source and field points that lie on the same radial distance from the axis of the cylinder such as the air dielectric and dielectric dielectric interfaces including the axial line rho rho and phi phi which has not been available before in the course of obtaining these expressions the conventional spectral domain green s function representations are rewritten in a different form so that i we can attack the axial line problem and ii the method can handle electrically large cylinders available acceleration techniques that exist in the literature are implemented to perform the summation over the cylindrical eigenmodes efficiently lastly the resulting expressions are transformed to the spatial domain using the discrete complex image method with the help of the generalized pencil of function method where a modified two level approach is used numerical results are presented in the form of mutual coupling between two current modes to assess the accuracy of the final spatial domain cfgf representations

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

focusing a mechanism for instability of nonlinear finite difference equations abstract a mechanism for the destabilization of numerical algorithms for partial differential equations is suggested the novelty of the work is that it attempts to explain the dynamical process by which noise can localize on a spatial grid and cause finite amplitude instability thresholds to be exceeded at distinct locations

parametric excitation of computational modes inherent to leap frog scheme applied to the korteweg de vries equation abstract the numerical instability in the usage of the leap frog scheme applied to the korteweg de vries equation is studied the leap frog scheme generates computational modes which are not included in the original partial differential equation it is shown that the numerical instability comes from the parametric excitation of the computational modes by physical waves a method to suppress the computational modes is given the method enables us to make long time integration of the korteweg de vries equation even if the amplitude of wave is large

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

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

gradient crystal plasticity as part of the computational modelling of polycrystals this paper describes how gradient hardening can in a thermodynamically consistent fashion be included into a crystal plasticity model by assuming that the inelastic part of the free energy includes contributions from the gradient of hardening along each slip direction a hardening stress due to the second derivative of the hardening along each slip direction can be derived for a finite element model of the grain structure a coupled problem with displacements and gradient hardening variables as degrees of freedom is thereby obtained this problem is solved using a dual mixed approach in particular an algorithm suitable for parallelization is presented where each grain is treated as a subproblem the numerical results show that the macroscopic strength increases with decreasing grain size as a result of gradient hardening finally the results of different prolongation assumptions i e how to impose the macroscopic deformation gradient on a representative volume element are compared

computational homogenization of mesoscale gradient viscoplasticity variationally consistent selective homogenization is applied to a class of gradient enhanced dissipative materials which is adopted for the mesoscale modeling the adopted standard first order homogenization assumption results in the classical equilibrium equation for a local continuum on the macroscale while the internal variables live on the mesoscale only among the issues considered in the paper we note i the variationally consistent setting of homogenization of gradient theory on the mesoscale ii the variational basis of the sve problem sve statistical volume element in the time incremental setting the sve problem is formulated for the classical boundary conditions dirichlet and neumann pertinent to the standard momentum balance as well as the micro momentum balance the weak format of the sve problem constitutes the stationarity condition of an incremental sve potential which represents an extension of the situation for a single phase continuum model the macroscale stress in a given time increment is derivable from an incremental macroscale pseudo elastic strain energy moreover the saddle point properties of the sve potential are shown to form the basis for establishing upper and lower bounds on the pseudo elastic strain energy bingham viscoplasticity with gradient enhanced hardening is chosen as the prototype model problem for the numerical evaluation the computed stress strain response relations confirm the theoretical predictions of the influence of different combinations of boundary conditions on the sve

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

variational gradient plasticity at finite strains part ii local global updates and mixed finite elements for additive plasticity in the logarithmic strain space the second part of our work on variational inelasticity with long range effects outlines the formulation and finite element implementation of additive finite gradient plasticity in the logarithmic strain space it is considered to be the most simple approach to finite plasticity suitable for the purely phenomenological description of polycrystalline metals or amorphous materials if structures of the geometrically linear theories are defined in the lagrangian logarithmic strain space we start from a mixed saddle point principle for metric type additive plasticity which is specified for the important model problem of isochoric von mises plasticity with gradient extended hardening softening response the mixed variational structure includes the hardening softening variable itself as well as its dual driving force the numerical implementation exploits the underlying variational structure yielding a canonical symmetric structure of the monolithic problem it results in a novel finite element design of the coupled problem incorporating a long range hardening softening parameter and its dual driving force this allows a straightforward local definition of plastic loading unloading driven by the long range fields providing very robust finite element implementations of gradient plasticity it includes a rational method for the definition of elastic plastic boundaries epbs in gradient plasticity along with a postprocessor that defines the plastic variables in the elastic range we discuss alternative mixed finite element designs of the coupled problem including a local global solution strategy of short and long range fields all methods are derived in a rigorous format from variational principles numerical benchmarks demonstrated the excellent performance of the proposed mixed variational approach to gradient plasticity in the logarithmic strain space

finite element analysis of geometrically necessary dislocations in crystal plasticity we present a finite element method for the analysis of ductile crystals whose energy depends on the density of geometrically necessary dislocations gnds we specifically focus on models in which the energy of the gnds is assumed to be proportional to the total variation of the slip strains in particular the gnd energy is homogeneous of degree one in the slip strains such models indeed arise from rigorous multiscale analysis as the macroscopic limit of discrete dislocation models or from phenomenological considerations such as a line tension approximation for the dislocation self energy the incorporation of internal variable gradients into the free energy of the system renders the constitutive model non local we show that an equivalent free energy functional which does not depend on internal variable gradients can be obtained by exploiting the variational definition of the total variation the reformulation of the free energy comes at the expense of auxiliary variational problems which can be efficiently solved using finite element approximations the addition of surface terms in the formulation of the free energy results in additional boundary conditions for the internal variables the proposed framework is verified by way of numerical convergence tests and simulations of three dimensional problems are presented to showcase its applicability a performance analysis shows that the proposed framework solves strain gradient plasticity problems in computing times of the order of local plasticity simulations making it a promising tool for non local crystal plasticity three dimensional large scale simulations

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

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

calculating the mutual coupling between apertures on a convex circular cylinder using a hybrid utd mom method the problem of calculating the mutual coupling between apertures on a perfectly conducting circular cylinder is considered the mutual coupling is calculated by a combination of the uniform theory of diffraction utd and the method of moments mom this hybrid method makes it possible to include higher order aperture modes in order to improve the accuracy it is found that the utd mom solution gives accurate results even for small cylinders the results are verified against results found in the literature as well as against measurements performed at ericsson microwave system in molndal sweden for this purpose a cylindrical test antenna with 54 apertures was built using a single mode approximation good agreement down to about 60 db with comparisons to measured data has been obtained if additional modes are added the results improve significantly both in amplitude and phase in several cases the agreement is good down to coupling levels of about 80 db this requires that about six modes are included however the most important mode besides the fundamental te sub 10 mode is the tm sub 11 mode

analysis of nearly cylindrical antennas and scattering problems using a spectrum of two dimensional solutions this paper presents a powerful method for analysing antennas which can be considered principally two dimensional 2 d or cylindrical except for some three dimensional 3 d physical or equivalent sources e g dipoles or slots it is shown by fourier transform techniques that such antennas can be analyzed as 2 d problems with harmonic longitudinal field variation the radiation pattern can often be determined directly from a finite set of such 2 d solutions each one obtained by any method e g the moment method the mutual interaction between the cylindrical scatterer and the sources must be calculated to determine the exact current distribution on the sources and their impedances or admittances this is facilitated by performing an inverse fourier transform of an infinite spectrum of the numerical 2 d solutions followed by a moment method solution in the spatial domain to satisfy the boundary conditions on the 3 d equivalent sources themselves the inverse fourier transform is simplified by the use of asymptote extraction the method is in itself a hybrid technique as one method is used to solve the harmonic 2 d problem and the other to solve for the source currents

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

a two field finite element formulation for elasticity coupled to damage a two field variational formulation is proposed for continuum damage mechanics problems this formulation is applied to the numerical solution via the finite element method of initial boundary value problems arising in elasticity coupled to damage the nodal variables are the displacement and damage fields simple examples are presented to demonstrate the advantages and limitations of this formulation

an anisotropic gradient damage model for quasi brittle materials an anisotropic continuum damage model based on the microplane concept is elaborated scalar damage laws are formulated on several individual microplanes representing the planes of potential failure these uniaxial constitutive laws can be cast into a fourthorder damage formulation such that anisotropy of the overall constitutive law is introduced in a natural fashion strain gradients are incorporated in the constitutive equations in order to account for microstructural interaction consequently the underlying boundary value problem remains well posed even in the softening regime the gradient continuum enhancement results in a set of additional partial diaerential equations which are satisfied in a weak form additional nodal degrees of freedom are introduced which leads to a modified element formulation the governing equations can be linearized consistently and solved within an incremental iterative newton raphson solution procedure the capability of the present model to properly simulate the localized failure of quasi brittle materials will be demonstrated by means of several examples 2000 elsevier science s a all rights reserved

the weak nonlinear instability of euler explicit scheme for the convective equation abstract the weak nonlinear instability of the euler explicit scheme for the convective equation u uux vxx is studied analytically using perturbation expansion and discrete fourier transform techniques short time weak nonlinear growth of the fourier modes for the noise is found provided that the linear stability conditions are satisfied as long as the perturbation expansion is valid the maximum growth is approximately proportional to the inverse of the grid spacing

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

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

strongly non local gradient enhanced finite strain elastoplasticity this paper presents a gradient plasticity formulation developed in a strongly non local format and applicable to ductile failure for finite strain plasticity the obtained formulation is an extension of a framework proposed by simo computer methods in applied mechanics and engineering 1988 66 199 into the softening and localization regime the presented model inherits the well established regularization properties of its strongly non local enrichment and leads to an implicit computational scheme with a consistent tangent operator numerical examples are given which illustrate the strongly non local character of the formulation the influence of the type of intrinsic length scale considered as well as several topics on plastic failure initiation and evolution into the geometrically non linear regime copyright 2003 john wiley sons ltd

analysis of a rectangular waveguide edge slot array with finite wall thickness a method to analyze a one dimensional array comprising tilted edge slots cut in the narrow wall of a rectangular waveguide is presented the fields in the slots are calculated from a hybrid finite element boundary integral fe bi equation method the boundary surface integrals on both the interior and exterior surface of the slots are computed from a spectrum of two dimensional solutions s2ds which allows the influence of both the interior and exterior structure of the waveguide to be included rigorously for initial array design the mutual coupling between slots in the exterior region is treated approximately with an infinite array assumption modeled by two slots surrounded by periodic boundary walls with an imposed linear phase progression a short finite slot array is designed based on the infinite array assumption and fabricated results from full wave analysis of the array and measurements are presented

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

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

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

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 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

on the comparison of two approaches to compute material forces for inelastic materials application to single slip crystal plasticity abstract the main goal of this contribution is the application of the material force method to dissipative materials whereby special emphasis is placed on the application to crystal plasticity balance equations on the material manifold as e g the material motion balance of momentum generally necessitate the computation of the spatial gradient of additional internal degrees of freedom or rather internal variables we therefore focus in this work on the comparison of two different numerical approaches to compute these gradient fields within a finite element setting namely a node point based and an integration point based formulation several numerical examples underline that the latter framework which is the convenient numerically comparatively cheap and well established approach in computational inelasticity turns out to be sufficiently accurate for the problem at hand

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

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

admittance of a longitudinal waveguide slot radiating into an arbitrary cylindrical structure in this paper we use a hybrid method to calculate the admittance of a longitudinal slot of finite length in a rectangular waveguide when the slot radiates into a cylindrical outer structure of arbitrary finite cross section the analysis takes advantage of a fourier transformation along the cylinder axis by which the computational effort is drastically reduced the harmonic spectral currents on the structure are expanded in subsectional triangular basis functions and determined using an electric field integral equation efie and the moment method these currents and the corresponding fields are inverse fourier transformed and matched over the slot to the fields inside the waveguide by an additional moment method approach using entire domain basis functions for the slot field the calculated slot admittance has good accuracy when compared with measurements on a chosen geometry