homogenization of two phase elasto plastic composite materials and structures study of tangent operators cyclic plasticity and numerical algorithms we develop homogenization schemes and numerical algorithms for two phase elasto plastic composite materials and structures a hill type incremental formulation enables the simulation of unloading and cyclic loadings it also allows to handle any rate independent model for each phase we study the crucial issue of tangent operators elasto plastic or continuum versus algorithmic or consistent and anisotropic versus isotropic we apply two methods of extraction of isotropic tangent moduli we compare mathematically the stiffnesses of various tangent operators all rate equations are discretized in time using implicit integration we implemented two homogenization schemes mori tanaka and a double inclusion model and two plasticity models classical j 2 plasticity and chaboche s model with non linear kinematic and isotropic hardenings we consider composites with different properties and present several discriminating numerical simulations in many cases the results are validated against finite element fe or experimental data we integrated our homogenization code into the fe program abaqus using a user material interface umat a two scale procedure allows to compute realistic structures made of non linear composite materials within reasonable cpu time and memory usage examples are shown c 2002 elsevier science ltd all rights reserved

computational modeling of deformation bands in granular media ii numerical simulations development of accurate mathematical models of geomaterial behavior requires a more fundamental understanding of the localization phenomena in particular the important factors responsible for the inception and development of localized deformation the objective of this paper is to implement and test the mathematical formulations presented in a companion paper to better understand the different failure processes in granular media specifically the formation of deformation bands in geomaterials our approach revolves around the use of classical bifurcation theory combined with advanced constitutive modeling and state of the art computation to capture the end members of the failures modes described in the companion paper namely simple shear pure compaction and pure dilation bands as well as the combination modes described in the geological framework the paper revisits the notion of the critical hardening modulus as it applies to the entire range of failure modes elucidates the role of the third stress invariant and finite deformation effects on the localization properties and describes some useful properties of the constitutive and algorithmic tangent operators as they relate to the capture of the onset of deformation bands in geomaterials

computational plasticity for composite structures based on mathematical homogenization theory and practice abstract this paper generalizes the classical mathematical homogenization theory for heterogeneous medium to account for eigenstrains starting from the double scale asymptotic expansion for the displacement and eigenstrain fields we derive a close form expression relating arbitrary eigenstrains to the mechanical fields in the phases the overall structural response is computed using an averaging scheme by which phase concentration factors are computed in the average sense for each micro constituent and history data is updated at two points reinforcement and matrix in the microstructure one for each phase macroscopic history data is stored in the database and then subjected in the post processing stage onto the unit cell in the critical locations for numerical examples considered the cpu time obtained by means of the two point averaging scheme with variational micro history recovery with 30 seconds on sparc 10 51 as opposed to 7 hours using classical mathematical homogenization theory at the same time the maximum error in the microstress field in the critical unit cell was only 3 5 in comparison with the classical mathematical homogenization theory

implicit algorithm for finite deformation hypoelastic viscoplasticity in fcc metals an implicit objective stress update algorithm is proposed for a hypoelastic viscoplastic model a thermal dynamic yield function which is derived based on the thermal activation analysis and dislocation interaction mechanisms is used along with the consistency approach and the framework of additive viscoplasticity in deriving the proposed model for fcc metals the corotational formulation approach is utilized in developing the proposed model in the finite deformation field for the case of the newton raphson iteration method a new expression for the consistent algorithmic tangent stiffness matrix of rate dependent metals is derived by direct linearization of the stress update algorithm finite element simulations are performed by implementing the proposed viscoplasticity constitutive models in the commercial finite element program abaqus numerical implementation for a simple tensile problem is used for validating the material parameters of the ofhc copper under low and high strain rates and temperatures the numerical results of the adiabatic true stress true strain curves compare very well with the experimental data the effectiveness of the present approach is tested by studying strain localization in a simple plane strain problem results indicate excellent performance of the present framework in describing the strain localization problem and in obtaining mesh independent results copyright 2006 john wiley sons ltd

finite deformation plasticity based on the additive split of the rate of deformation and hyperelasticity abstract finite deformation plasticity formulation based on additive split of rate of deformation and hyperelasticity is presented this approach is valid for finite elastic and plastic strains while rendering the choice and numerical integration of objective stress rates superfluous as the results are automatically objective for small elastic strains our method reduces to the classical hypoelastic corotational formulation provided that the dienes objective stress rate is employed while in the absence of inelastic deformation it coincides with the hyper elastic formulation the validity of the model has been examined on four test problems and the numerical results were found to be in good agreement with either the exact solution or experimental data

design sensitivity analysis for rate independent elastoplasticity abstract a new incremental direct differentiation method for design sensitivity analysis of structures with rate independent elastoplastic behavior is presented we formulate analytical sensitivity expressions that are consistent with numerical algorithms for elastoplasticity that use implicit methods to integrate the constitutive equations and return mappings to enforce the consistency conditions the sensitivity expressions can be evaluated with only a modest increase in computational expense beyond the cost of simulation combined with the inherent advantages of implicit integration strategies this represents a significant improvement over previous sensitivity formulations for history dependent materials first order sensitivity expressions involving the complete set of design variables including shape design variables are derived for a generic response functional the reduced form of the consistent tangent stiffness matrix obtained at the end of each time or load step in the finite element procedure is used to update the response sensitivities for that time step no iterations are needed in the sensitivity computations a numerical example demonstrates the accuracy and efficiency of the new sensitivity analysis method for an elastoplastic analysis problem explicit sensitivities from the new method are confirmed by finite difference estimates

analysis of the anisotropic viscoplastic damage response of composite laminates continuum basis and computational algorithms the mathematical structure underlying the rate equations of a recently developed constitutive model for the coupled viscoplastic damage response of anisotropic composites is critically examined in this regard a number of tensor projection operators have been identified and their properties were exploited to enable the development of a general computational framework for their numerical implementation using the euler fully implicit integration method in particular this facilitated i the derivation of explicit expressions of the consistent material tangent stiffnesses that are valid for both three dimensional as well as subspace e g plane stress formulations ii the implications of the symmetry or unsymmetry properties of these tangent operators from a thermodynamic standpoint and iii the development of an effective time step control strategy to ensure accuracy and convergence of the solution in addition the special limiting case of inviscid elastoplasticity is treated the results of several numerical simulations are given to demonstrate the effectiveness of the schemes developed

scaling of constitutive relationships in unsaturated heterogeneous media a numerical investigation numerical models of groundwater flow in the unsaturated zone require relative conductivity and moisture retention curves at a discretization scale much larger than that at which these constitutive relationships are measured this study seeks to identify simple scaling laws linking parameters of block scale constitutive relationships with the parameters of their measurement scale counterparts represented here using gardner russo models the two parameters of these models saturated conductivity and pore size distribution parameter are viewed as cross correlated regionalized variables with isotropic or anisotropic spatial correlation structures and are simulated using the sequential gaussian method upscaled constitutive curves for steady gravity drainage through three dimensional heterogeneous domains are then constructed by repeating numerical flow experiments for a series of applied infiltration rates and plotting the resulting upscaled relative conductivities and average water saturations against average matric potentials the experiments reveal clear log linear relationships between upscaled relative conductivities and average matric potentials the slopes of which are used to define an upscaled pore size distribution parameter the reciprocal of this quantity which represents an upscaled characteristic capillary length is very well approximated by a spatial power average of its measurement scale values furthermore the averaging exponent is found to be the same as that associated with the upscaling of saturated conductivities over the same flow domains when average matric potentials are normalized using this upscaled characteristic capillary length relative conductivity and moisture saturation values for widely different flow field realizations collapse along well defined curves however for wet conditions the form of these curves differs from that of the models assumed at the measurement scale

micromechanics of inelastic composites with misaligned inclusions numerical treatment of orientation this paper is concerned with the micromechanics of multi phase fiber or inclusion reinforced inelastic composites with a special emphasis on the numerical treatment of misaligned orientation mean field homogenization of these composites involves volume and orientation averaging the latter usually needs average orientation measures known as the second and fourth rank orientation tensors a and a respectively usually the only orientation data available is a and a first issue is to reconstruct a using closure approximations we propose a new weighted regularization method which deals with orientations which are strictly neither 1d 2d nor 3d and ensures a smooth transition between the 3 cases a second issue is that there are situations where the inclusions orientation distribution function odf is needed for homogenization but is unavailable the odf is recovered using a and a and numerically computed at a number of discrete orientations a third issue is the actual computation of orientation averaging integrals an efficient algorithm approximates the integrals with sums over a finite number of orientations orientation and volume averaging concepts are applied to the mean field homogenization of two classes of multi phase composites linear thermo elastic and rate independent inelastic in the latter case an incremental formulation is proposed which enables the simulation of unloading cyclic and otherwise non proportional loadings implicit time discretization and consistent tangent operators are employed the procedures and the corresponding algorithms were implemented in the digimat version 1 4 2004 linear and nonlinear multi scale material modeling software e xstream engineering sa http www e xstream com louvain la neuve belgium software and several numerical simulations show their accuracy and efficiency c 2005 elsevier b v all rights reserved

stochastic modeling of large scale flow in heterogeneous unsaturated soils a simulation experiment is used to test the stochastic theory of unsaturated flow proposed by mantoglou and gelhar 1987a b c tension means and variances derived from this theory are compared to tension distributions obtained from a detailed three dimensional model developed by ababou 1988 the synthetically generated unsaturated hydraulic conductivity functions used in the detailed model vary significantly over relatively small distances results from a simulated infiltration event indicate that the stochastic theory s predictions reproduce the overall trend of the heterogeneous tension distribution deviations from the predicted mean generally lie within confidence intervals derived from the predicted variance the stochastic theory predicts less vertical moisture movement and somewhat more horizontal spreading than a comparable deterministic analysis this reflects the fact that the effective hydraulic conductivity function used in the stochastic approach is anisotropic the magnitude of this anisotropy increases with increasing tension

on the implementation of inelastic constitutive equations with special reference to large deformation problems abstract methods for finite element analysis of inelastic problems are reviewed problems with currently popular methods based on rate principles of virtual work are pointed out a formulation is presented based on the exact incremental virtual work equation the procedure for integration of the constitutive rate equations to obtain incremental constitutive equations is based on the assumption of a constant strain rate or straight strain path during the increment full newton raphson iteration is used to solve the resulting set of nonlinear equations the treatment of the constitutive equations in this newton raphson scheme leads to a linearized form which is substantially different from the classical rate equations the procedure is implemented in a finite element program and its effectiveness is shown in a number of examples

consistent derivation of the constitutive algorithm for plane stress isotropic plasticity part i theoretical formulation abstract a derivation of the projected algorithm for general isotropic three invariant plasticity models under plane stress conditions is presented it is obtained by consistently specializing the 3d formulation to the 2d subspace defined by the plane stress condition closed form intrinsic algorithm linearization and a novel expression of the consistent tangent tensor are provided these are also shown to directly emanate from the analogous quantities pertaining to the fully 3d case a detailed discussion of the proposed implementation along with a representative set of numerical examples is provided in the second part of this paper valoroso n rosati l 2008 consistent derivation of the constitutive algorithm for plane stress isotropic plasticity part ii computational issues international journal of solids and structures 46 92 124

finite deformation post buckling analysis involving inelasticity and contact constraints this paper is concerned with the numerical solution of large deflection structural problems involving finite strains subject to contact constraints and unilateral boundary conditions and exhibiting inelastic constitutive response first a three dimensional finite strain beam model is summarized and its numerical implementation in the two dimensional case is discussed next a penalty formulation for the solution of contact problems is presented and the correct expression for consistent tangent matrix is developed finally basic strategies for tracing limit points are reviewed and a modification of the arc length method is proposed the good performance of the procedures discussed is illustrated by means of numerical examples

a return mapping algorithm for a class of cyclic plasticity models a return mapping algorithm for a class of cyclic plasticity models is presented the constitutive model includes a quadratic yield criterion and multi component formulations of non linear isotropic and kinematic hardening thus both initially isotropic and anisotropic materials can be described and a proper description of the hysteresis loops and the cyclic hardening response of materials under cyclic loading is obtained the return mapping algorithm employs the closest point projection scheme in combination with a decomposition method to obtain an efficient and robust integration of the constitutive model the consistent tangent operator is derived in closed form and is found to be unsymmetric due to the non linear evolution of the kinematic hardening terms the accuracy and robustness of the algorithm are assessed through numerical examples

algorithmic issues for three invariant hyperplastic critical state models implicit stress integration and the consistent tangents are presented for critical state hyperplasticity models which include a dependence on the third invariant of stress an elliptical deviatoric yielding criterion 43 is incorporated within the family of geotechnical models first proposed by collins and hilder 8 an alternative expression for the yield function is proposed and the consequences of different forms of that function are revealed in terms of the stability and efficiency of the stress return algorithm errors associated with the integration scheme are presented it is shown how calibration of the two new material constants is achieved through examining one dimesional consolidation tests and undrained triaxial compression data material point simulations of drained triaxial compression tests are then compared with established experimental results strain probe analyses are used to demonstrate the concepts of energy dissipation and stored plastic work along with the robustness of the integration method over twenty finite element boundary value problems are then simulated these include single three dimensional element tests plane strain footing analyses and cavity expansion tests the rapid convergence of the global newton raphson procedure using the consistent tangent is demonstrated in small strain and finite deformation simulations

estimation of finite difference interblock conductivities for simulation of infiltration into initially dry soils simulations of transient unsaturated vertical flow in different initially dry homogeneous soils using richards equation suggest that for a relatively coarse numerical grid there is no universal weighting scheme to estimate interblock conductivities capable of giving an accurate resolution of the resulting steep wetting fronts except for the scheme associated with the kirchhoff transformation kt the principal drawback of kt is that it can hardly be applied to heterogeneous soils a new asymptotic weighting aw scheme based on the asymptotic behavior of the hydraulic conductivity function in the vicinity of residual saturation is proposed numerical tests demonstrate that the aw scheme describes wetting fronts in homogeneous soils nearly as accurately as a scheme based on kt applicability and robustness of the aw scheme for simulating water infiltration into heterogeneous soils having substantial vertical variation of hydraulic properties are also demonstrated extension of the proposed scheme to higher dimensions is straightforward

efficient and modular algorithms in modeling finite inelastic deformations objective integration parameter identification and sub stepping techniques this paper presents a general algorithmic formulation for constitutive equations describing inelastic material deformation the algorithmic representation focuses on the generality and modularity which is a key aspect within the development phase of constitutive equations it is shown explicitly how to solve the incremental constitutive equations by use of objective algorithms and an implicit integration scheme applying numerical as well as analytical derivatives although it is shown that using numerical gradients is significantly slower their application is easier especially in the development phase it is also demonstrated that having the constitutive equations modeled in the general context introduced the issue of parameter identification can be tackled by the same implementation using a driver algorithm besides parameter identification the general structure is also applied to two different fe simulations first the objectivity of different integration methods will be presented considering a simple shear loading superposed by a rigid rotation using a strain rate independent material model as example in the second example it is shown for a strain rate dependent material model how the simulation of the wire drawing process can be accelerated crucially by using sub stepping techniques

implicit bem formulations for usual and sensitivity problems in elasto plasticity using the consistent tangent operator concept this paper presents boundary element method bem formulations for usual and sensitivity problems in small strain elasto plasticity using the concept of the local consistent tangent operator cto usual problems here refer to analysis of nonlinear problems in structural and solid continua for which simo and taylor first proposed the use of the cto within the context of the finite element method fem a new implicit bem scheme for such problems using the cto is presented first a formulation for sensitivity analysis follows it is shown that the sensitivity of the strain increment associated with an infinitesimal variation of some design parameter solves a linear problem which is governed by the converged value of the same global cto as the one that appears in the usual problem numerical results for both usual and sensitivity problems are shown for a one dimensional example they demonstrate the effectiveness of the present approach in particular accurate sensitivities with respect to material parameters e g exponent of the power type hardening law are obtained even with few integration cells and for large load increments

a nonsmooth newton method for elastoplastic problems in this work we reformulate the incremental small strain j2 plasticity problem with linear kinematic and nonlinear isotropic hardening as a set of unconstrained nonsmooth equations the reformul

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

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

a formulation of the mitc4 shell element for finite strain elasto plastic analysis abstract a finite strain elasto plastic formulation is developed for the mitc4 shell element the developed formulation is based on lee s multiplicative decomposition of the deformation gradient and on the hyperelastic expression of von mises flow rule expressed in terms of hencky s strain tensor the formulation incorporates thickness stretching degrees of freedom that are condensed at the element level by imposing the classical in layer plane stress condition a symmetric and consistent tangent stiffness matrix is also developed

an efficient method for simulating steady unsaturated flow in random porous media using an analytical perturbation solution as initial guess to a numerical model numerical simulation of flow through multidimensional heterogeneous soils under unsaturated conditions is a computationally intensive task the governing unsaturated flow equation is nonlinear the degree of nonlinearity depends on the unsaturated hydraulic properties of the soil and the degree of heterogeneity in this paper we present a highly efficient method to simulate unsaturated steady state flow through random porous media our method utilizes the analytical approximate solution derived by a perturbation spectral method as an initial guess solution for a numerical model to simulate two dimensional vertical infiltration problems it is found that this approach which we call asigning reduces the required cpu time by one to two orders of magnitude asigning is demonstrated to operate successfully under a wide variety of boundary conditions which may substantially deviate from those imposed on the initial guess solution a large range of mean and variances in the independent variables ln ks and or alternatively in has been tested and it is shown that the method works well for variances of the unsaturated hydraulic conductivity ln2k 5 and average 0 1 cm 1

large elasto plastic finite element analysis of solids and shells with the enhanced assumed strain concept abstract a finite element simulation of arbitrarily large elasto plastic deformations of three dimensional and shell structures is described two numerical aspects originally developed by j c simo have been combined i e the multiplicative large strain plasticity and the enhanced assumed strain finite element concept the constitutive model is worked out in its algorithmic version applying projection methods and consistent linearization the enhanced strain field is added to the conventional hexahedral displacement elements avoiding volumetric locking in the plastic range in addition to solid structures the three dimensional material model without any modifications is also applied to thin walled shell structures here an extended shell formulation a so called seven parameter model is used in which the thickness strain is augmented by a linear term the applicability of the entire formulation is demonstrated by several numerical examples

an integration scheme for prandtl reuss elastoplastic constitutive equations we investigate the generalized midpoint rule for the time integration of elastoplastic constitutive equations for pressure independent yield criteria the incremental equations are divided into one scalar hydrostatic pressure dilation rate equation and a stress deviator strain rate deviator tensorial equation this formulation is applied to two classical problems bulging of a tube under internal pressure and tension test on a notched specimen

solution of the slump test using a finite deformation elasto plastic drucker prager model summary development of a finite deformation elasto plasticity model based on the multiplicative decomposition of the deformation gradient is presented and discussed in detail the formulation presented in this paper includes the derivation of the full set of equations for the drucker prager yield criterion the equations which are not available elsewhere are developed within a framework using a spectral decomposition approach further expressions for the consistent algorithmic tangent moduli in the finite strain regime are developed since the finite deformation framework employed to obtain the expressions presented here collapses to the classical infinitesimal plasticity framework when the finite strain assumption is no longer necessary the finite deformation consistent tangent moduli are compared to the consistent tangent moduli valid for use with infinitesimal plasticity validation of the implemented finite deformation elasto plastic drucker prager model is performed through the solution of the concrete slump test comparisons between an existing approximate analytical solution and experimental data are presented and results are discussed in detail

a four point integration scheme with quadratic exactness for three dimensional element free galerkin method based on variationally consistent formulation abstract the formulation of three dimensional element free galerkin efg method based on the hu washizu three field variational principle is described the orthogonality condition between stress and strain difference is satisfied by correcting the derivatives of the nodal shape functions this leads to a variationally consistent formulation which has a similar form as the formulation of standard galerkin weak form based on this formulation an integration scheme which employs only four cubature points in each background tetrahedral element cell is rationally developed for three dimensional efg with quadratic approximation the consistency of the corrected nodal derivatives and the satisfaction of patch test conditions for the developed integration scheme are theoretically proved extension of the proposed method to small strain elastoplasticity is also presented the proposed method can exactly pass quadratic patch test that is quadratic exactness is achieved and thus it is named as quadratically consistent 4 point qc4 integration method in contrast efg with standard tetrahedral cubature and the existing linearly consistent 1 point lc1 integration fail to exactly pass quadratic patch test numerical results of elastic examples demonstrate the superiority of the proposed method in accuracy convergence efficiency and stability the capability of the proposed qc4 scheme in solving elastoplastic problems is also demonstrated by numerical examples

an interblock conductivity scheme for finite difference models of steady unsaturated flow in heterogeneous media with the increased ability to model subsurface heterogeneity provided by stochastic simulation methods there is a corresponding increase in the need for numerical flow models to accomodate the spatial variability of hydrodynamic parameters these trends make it necessary to reexamine methods for calculating finite difference interblock conductivities originally developed for use in homogeneous media this study describes a new method for calculating interblock conductivities better suited to the numerical modeling of unsaturated flow in heterogeneous media an expression for interblock conductivity is derived by equating the finite difference flux between adjacent blocks with the exact analytical flux for one dimensional steady unsaturated flow assuming an exponential model for relative conductivity this expression appears as a function of the saturated conductivities alpha parameters and matric potentials of the adjacent blocks and the potential at the interblock boundary since this last quantity is not directly featured in the discretized flow equation it must be evaluated iteratively the interblock conductivity calculation scheme is implemented within a block centered seven point finite difference model of steady unsaturated flow where nonlinearity is resolved using a newton raphson iterative approach in a series of numerical experiments this scheme is compared with the more familiar geometric averaging method the proposed interblock conductivity scheme is shown to provide perfect agreement between numerical and analytical solutions for the one dimensional case of steady infiltration into a perfectly layered medium in problems of two dimensional unsaturated flow in heterogeneous media comparisons with the geometric averaging method show that the proposed scheme allows more rapid convergence of newton raphson iterations and consideration of greater parameter spatial variability before the onset of divergence in these iterations

finite elastoplastic deformations of space curved membranes abstract under restriction of an isotropic elastic response of deformed lattice a covariant theory of finite elastoplasticity is developed in principal axes of a pair of deformation tensors in material description the tensor pair consists of the plastic deformation tensor and the total deformation cauchy green tensor the proposed theory is applied to elastoplastic membranes whose reference and current configurations can be arbitrary space curved surfaces pressure insensitive von mises yield criterion and isotropic hardening are considered as a model problem with a particular form of the strain energy function given in terms of elastic principal stretches through an explicit enforcement of the plane stress condition we arrive at a reduced two dimensional problem representation which is set in the membrane tangent plane numerical implementation details are given to show an important role of the operator split methodology in simplifying the state update computation and the computation of the consistent tangent modulus a set of numerical examples illustrates the performance of the presented theory and indicates some of the possible areas of application

design sensitivity analysis for parameters affecting geometry elastic viscoplastic material constant and boundary condition by consistent tangent operator based boundary element method abstract this paper presents a design sensitivity analysis method by the consistent tangent operator concept based boundary element implicit algorithm the design variables for sensitivity analysis include geometry parameters elastic viscoplastic material parameters and boundary condition parameters based on small strain theory perzyna s elastic viscoplastic material constitutive relation with a mixed hardening model and two flow functions is considered in the sensitivity analysis the related elastic viscoplastic radial return algorithm and the formula of elastic viscoplastic consistent tangent operator are derived and discussed based on the direct differentiation approach the incremental boundary integral equations and related algorithms for both geometric and elastic viscoplastic sensitivity analysis are developed a 2d boundary element program for geometry sensitivity elastic viscoplastic material constant sensitivity and boundary condition sensitivity has been developed comparison and discussion with the results of this paper analytical solution and finite element code ansys for four plane strain numerical examples are presented finally

uniqueness loss of ellipticity and localization for the time discretized rate dependent boundary value problem with softening the discretization in time of the initial boundary value problem for rate dependent elastic viscoplastic solid materials in presence of softening is investigated in this paper the emphasis is put on uniqueness loss of ellipticity and localization it is found that the time discretized problem resembles the incremental problem for rate independent materials and softening may lead to ill posedness loss of ellipticity if the time step is greater than a critical value it is well established that the implication of loss of ellipticity for the numerical simulations after spatial discretization is the pathological mesh dependency of the computed results we furnish a method to compute the critical time step and demonstrate its use for a simple example problem copyright 2010 john wiley sons ltd

numerical aspects in the finite element simulation of the portevin le chatelier effect abstract serrated yielding and propagation of localized bands of plastic strain rate are the apparent phenomena of the portevin le chatelier plc effect the finite element modeling of this effect is investigated using a model proposed by zhang et al 74 describing dynamic strain aging and material parameters for a nickel based superalloy at 500 c this work presents 1 an efficient implicit integration scheme of the constitutive equations in the presence of instabilities 2 a numerical tool to determine the type of plastic strain rate localization bands based on results of simulations and 3 a mesh and time discretization sensitivity analysis of the model regarding the characteristics of plc bands this analysis is carried out in 2d and 3d for axisymmetric smooth and notched specimens

coupled plastic damaged model abstract a constitutive model that couples plasticity and damage is presented the model is thermodynamically consistent and comes from a generalization of classical plasticity theory and isotropic damage theory of kachanov coupling between plasticity and damage is achieved through a simultaneous solution of the plastic and the damage problem after a description of the model a numerical algorithm for the integration of the resulting constitutive equations is presented it is an euler backward type of algorithm that is particularly suitable to solve plain stress non linear problems with a 2d finite element program the consistent stiffness matrix is also derived the paper is completed with some application examples that show that the model presented accurately reproduces the behaviour of elastic plastic damaged materials

elastoplastic materials with martensitic phase transition and twinning at finite strains numerical solution with the finite element method abstract a new quasistatic problem formulation and finite element fe algorithm for martensitic phase transition pt and twinning in elastoplastic materials at large strains based on a recently proposed thermomechanical approach 1 2 are presented the instantaneous occurrence of pt in some region based on thermodynamics without introduction of volume fraction and prescribing the kinetic equations for it is considered stress history dependence during the transformation process is a characteristic feature of the pt criterion the deformation model is based on the multiplicative decomposition of the total deformation gradient into elastic transformation and plastic parts and the generalization of prandtl reuss equations to the case of large strains and pt the case of small elastic but large plastic and transformation strains is assumed for numerical simulation of pt the inverse problem is considered i e the position and size of the pt region nucleus are prescribed in advance and then the condition for pt is defined from the pt criterion such an approach includes a finite element solution of the elastoplastic problem with the prescribed transformation deformation gradient and the changing elastoplastic properties in the transforming region during pt the usage of the current configuration and the true cauchy stresses along with assumptions of small elastic strains and zero modified plastic spin allows us to use with small modifications the radial return algorithm and the consistent elastoplastic moduli for the case of small strains some modifications of the iterative algorithm related to the numerical integration of constitutive equations along with the radial return algorithm are suggested in order to improve the accuracy of solutions for large increments of external load such modifications can be used for any elastoplastic problem without phase transition as well the model problems of nucleation at shear band intersection and appearance of a single martensitic plate and a single twin are solved and analyzed

a finite element constitutive update scheme for anisotropic viscoelastic solids exhibiting non linearity of the schapery type this study presents a new scheme for performing integration point constitutive updates for anisotropic small strain non linear viscoelasticity within the context of implicit non linear finite element structural analysis while the basic scheme has been presented earlier by the authors for linear viscoelasticity the present work illustrates the generality of the underlying fundamentals by extending to schapery s non linear model the method features a judicious choice of state variables a stable backward euler integration step and a consistent tangent operator its greatest strength lies in ready incorporation into existing fem codes numerical examples involving homogeneous stress states such as uniaxial extension and simple shear and non uniform stress states such as a beam under tip load were carried out by incorporating the present scheme into a general purpose fem package excellent agreement with analytical results is observed copyright 1999 john wiley sons ltd

nonlinear thin shell analysis using a curved triangular element abstract this paper describes a simple but robust curved triangular thin shell element the element has twelve degrees of freedom namely the displacement components in the corner nodes and the rotations about the element sides the element is based upon the well known facet element with constant membrane strains and constant stress couples and is augmented to take into account the influence of the changes of curvature on the membrane deformations due to the quadratic character of this influence curved geometries can be modelled in order to permit arbitrarily large rotational increments reference vectors are attached to each element side these reference vectors are updated by means of orthogonal transformations the latter are determined uniquely by the subsequent orientations of the associated element sides the changes of curvature are evaluated using these reference vectors and a corotated reference configuration to describe time independent elastoplasticity a layered shell model is applied while besseling s fraction model is adopted for each layer a number of numerical examples illustrate the performance of the element

fully implicit formulation of elastoplastic homogenization problem for two scale analysis abstract in this study using a virtual work equation a micro macro kinematic relation and a linearized constitutive relation a boundary value problem is fully implicitly formulated to determine perturbed displacement increment fields in elastoplastic unit cells for two scale analysis it is shown that this implicit homogenization problem can be iteratively solved with quadratic convergences by successively updating strain increment fields in unit cells and that the boundary value problem formulated provides a computational algorithm which is versatile for initial setting of strain increment fields the computational algorithm developed is then examined by performing a two scale analysis of a holed plate with an elastoplastic micro structure subjected to tensile loading this demonstrates that the convergence in iteratively solving the implicit homogenization problem strongly depends on the initial setting of strain increment fields in unit cells

design sensitivity analysis of elastoplastic structures an iterative and incremental algorithm is presented for the design sensitivity analysis dsa of elastoplastic structures geometrical non linearity is included in the formulation and the structure may be subjected to a cyclic loading the key concept is discussed using an elastoplastic truss it is shown that the design sensitivity coefficients dscs are path dependent and discontinuous at the time at which yielding takes place in a member yield time in the proposed algorithm the yield time is considered to be a function of the design variables in this way the discontinuity in the dscs at the material transition points can be overcome in a simple and routine way incremental response analysis and dsa are carried out simultaneously using the newton raphson type iterative procedure since the proposed algorithm is completely consistent with the analysis procedure it can be implemented into an existing code for structural analysis application to distributed parameter structures with kinematic and or isotropic hardening is discussed using the von mises yield condition and the elastic predictor radial return method dsa is carried out for a ten bar truss with a piecewise linear constitutive relation it is shown that the dscs found using the proposed method agree quite well with those calculated by the central difference method finally an alternate method without any iteration for dsa is proposed and the results from the two methods are compared

an inverse micro mechanical analysis toward the stochastic homogenization of nonlinear random composites abstract an inverse mean field homogenization mfh process is developed to improve the computational efficiency of non linear stochastic multiscale analyzes by relying on a micro mechanics model first full field simulations of composite stochastic volume element sve realizations are performed to characterize the homogenized stochastic behavior the uncertainties observed in the non linear homogenized response which result from the uncertainties of their micro structures are then translated to an incremental secant mfh formulation by defining the mfh input parameters as random effective properties these effective input parameters which correspond to the micro structure geometrical information and to the material phases model parameters are identified by conducting an inverse analysis from the full field homogenized responses compared to the direct finite element analyzes on sves the resulting stochastic mfh process reduces not only the computational cost but also the order of uncertain parameters in the composite micro structures leading to a stochastic mean field reduced order model mf rom a data driven stochastic model is then built in order to generate the random effective properties under the form of a random field used as entry for the stochastic mf rom embedded in a stochastic finite element method sfem the two cases of elastic unidirectional ud fibers embedded in an elasto plastic matrix and of elastic ud fibers embedded in a damage enhanced elasto plastic matrix are successively considered in order to illustrate the capabilities of the method the stochastic response of a ply is studied under transverse loading condition

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

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

a generalized approach to formulate the consistent tangent stiffness in plasticity with application to the gld porous material model abstract it has been shown that the use of the consistent tangent moduli is crucial for preserving the quadratic convergence rate of the global newton iterations in the solution of the incremental problem in this paper we present a general method to formulate the consistent tangent stiffness for plasticity the robustness and efficiency of the proposed approach are examined by applying it to the isotropic material with j2 flow plasticity and comparing the performance and the analysis results with the original implementation in the commercial finite element program abaqus the proposed approach is then applied to an anisotropic porous plasticity model the gologanu leblond devaux model performance comparison between the consistent tangent stiffness and the conventional continuum tangent stiffness demonstrates significant improvement in convergence characteristics of the overall newton iterations caused by using the consistent tangent matrix

on the numerical integration of a class of pressure dependent plasticity models with mixed hardening radial return algorithms for both three dimensional and plane stress situations are developed for a class of pressure dependent plasticity models formulated in state variables with mixed hardening the consistent tangent matrix has been developed which among other advantages does not require numerical inversion the algorithms for gurson s mixed hardening model are incorporated in a finite element program to solve several simple uniaxial tension problems when compared with numerically integrated solutions the results from the finite element analysis show excellent agreement finally results of the axisymmetric necking problem using gurson s mixed hardening model are presented as an example of application

on the representation and implicit integration of general isotropic elastoplasticity based on a set of mutually orthogonal unit basis tensors summary a simple and compact representation framework and the corresponding efficient numerical integration algorithm are developed for constitutive equations of isotropic elastoplasticity central to this work is the utilization of a set of mutually orthogonal unit tensor bases and the corresponding invariants the set of bases can be regarded equivalently as a local cylindrical coordinate system in the three dimensional coaxial tensor subspace namely the principal space the base tensors are given in the global coordinate system similar to the principal space approach the proposed method reduces the problem dimension from six to three in contrast to the conventional approach the transformation procedure between the principal space and the general space is avoided and explicit computation of the principal axes is bypassed with the proposed technique the matrices which need to be inverted during iteration take a simple form for the great majority of constitutive equations in use the tangent operator consistent with the proposed algorithm can be decomposed into the direct sum of two linear maps over the coaxial tensor subspace and the subspace orthogonal to it consequently its closed form is derived in an extremely simple manner finally numerical examples demonstrate the high quality performances of the proposed scheme copyright 2014 john wiley sons ltd

a comparison of picard and newton iteration in the numerical solution of multidimensional variably saturated flow problems picard iteration is a widely used procedure for solving the nonlinear equation governing flow in variably saturated porous media the method is simple to code and computationally cheap but has been known to fail or converge slowly the newton method is more complex and expensive on a per iteration basis than picard and as such has not received very much attention its robustness and higher rate of convergence however make it an attractive alternative to the picard method particularly for strongly nonlinear problems in this paper the picard and newton schemes are implemented and compared in one two and three dimensional finite element simulations involving both steady state and transient flow the eight test cases presented highlight different aspects of the performance of the two iterative methods and the different factors that can affect their convergence and efficiency including problem size spatial and temporal discretization initial solution estimates convergence error norm mass lumping time weighting conductivity and moisture content characteristics boundary conditions seepage faces and the extent of fully saturated zones in the soil previous strategies for enhancing the performance of the picard and newton schemes are revisited and new ones are suggested the strategies include chord slope approximations for the derivatives of the characteristic equations relaxing convergence requirements along seepage faces dynamic time step control nonlinear relaxation and a mixed picard newton approach the tests show that the picard or relaxed picard schemes are often adequate for solving richards equation but that in cases where these fail to converge or converge slowly the newton method should be used the mixed picard newton approach can effectively overcome the newton scheme s sensitivity to initial solution estimates while comparatively poor performance is reported for the various chord slope approximations finally given the reliability and efficiency of current conjugate gradient like methods for solving linear nonsymmetric systems the only real drawback of using newton rather than picard iteration is the algebraic complexity and computational cost of assembling the derivative terms of the jacobian matrix and it is suggested that both methods can be effectively implemented and used in numerical models of richards equation

numerical approximations of darcian flow through unsaturated soil the choice of appropriate intrablock approximations for the unsaturated hydraulic conductivity is a critical step for the numerical solution of richards equation forms commonly used include arithmetic geometric and harmonic averaging the choice is most often based on convenience and custom this study presents a framework for comparing any scheme for weighting the unsaturated hydraulic conductivity against the correct darcian flow across a block for steady conditions such comparisons are carried out for a number of commonly used averages and errors are shown to be alarmingly high in some cases a scheme for approximately matching the correct darcian flux is illustrated by two infiltration examples in the first the arithmetic and weighted schemes are superior to the geometric average when compared to a quasi analytical solution for the second example the weighted solution is slightly better than either the arithmetic or geometric averaging schemes these examples also illustrate that an acceptable mass balance should not be used as a sole criterion for success as it can probably be attained with most schemes the conclusions are believed relevant for algorithms which are pressure head based water content based or other choices and for multidimensional as well as one dimensional flow problems

finite deformation formulation for embedded frictional crack with the extended finite element method we reformulate an extended finite element fe framework for embedded frictional cracks in elastoplastic solids to accommodate finite deformation including finite stretching and rotation for the fe representation we consider a galerkin approximation in which both the trial and weighting functions adapt to the current contact configuration contact and frictional constraints employ two kuhn tucker conditions a contact separation constraint nesting over a stick slip constraint for the case when the crack faces are in frictional sliding mode we integrate finite deformation bulk plasticity into the formulation using the multiplicative decomposition technique of nonlinear continuum mechanics we then present plane strain simulations demonstrating various aspects of the extended fe solutions the mechanisms considered include combined opening and frictional sliding in initially straight curved and s shaped cracks with and without bulk plasticity to gain further insight into the extended fe solutions we perform mesh convergence studies focusing on both the global and the local responses of structures with cracks including the distribution of the normal component of traction on the crack faces copyright 2009 john wiley sons ltd

material growth in thermoelastic continua theory algorithmics and simulation a theoretical formulation of stress induced volumetric material growth in thermoelastic continua is developed material growth is regulated by a three surface activation criterion and corresponding flow rules a simple model is proposed based on this formulation and its algorithmic implementation including a method for solving the underlying differential algebraic equations for growth is examined in the context of an implicit finite element method selected numerical simulations are presented that showcase the predictive capacity of the model for both soft and hard biomaterials

computational model for 3 d contact problems with friction based on the penalty method the friction forces are assumed to follow the coulomb law with a slip criterion treated in the context of a standard return mapping algorithm consistent linearization of the field equations is performed which leads to a fully implicit scheme with non symmetric tangent stiffness which preserves asymptotic quadratic convergence of the newton raphson method

accurate stress updating algorithm based on constant strain rate assumption abstract the calculation of the stresses for path dependent materials possesses a delicate problem here the constitutive equations are reformulated using the assumption of constant strain rate within an increment this assumption allows the constitutive laws to be rewritten as ordinary differential equations and thus a large set of tools is available to solve the constitutive equations within this concept the possibility to form an algorithmic stiffness tensor is investigated and it turns out that the true tensor is expensive to calculate but it is of interest that an inexpensive well behaving approximative algorithmic stiffness tensor can be formulated to investigate the properties of the solution method both with respect to stress calculation and convergence properties in a newton raphson method an elasto plastic damage model based on von mises plasticity is considered an interesting observation is that for the usual closest point projection method the damage evolution is very sensitive to the size of load step

numerical simulation of ductile fracture with the rousselier constitutive law this contribution aims at developing robust and reliable tools dedicated to predictive numerical simulation of ductile fracture description of damage mechanisms relies here on the rousselier constitutive law to achieve robustness and objectivity several points need then to be addressed reduction of the set of internal variables so as to recast the model into the framework of generalised standard materials proper expression of the normal plastic flow rule at the vertex of the yield surface finite strain formulation based on a multiplicative split of the deformation gradient fully implicit integration of the constitutive equations leading to a single scalar equation that admits a unique root use of mixed finite elements to avoid volumetric locking strain gradient based nonlocal model to control strain localisation and remedy spurious mesh dependency the capabilities of the resulting numerical formulation are demonstrated through simulating the cup cone fracture of a notched tensile specimen

internal symmetry groups for the drucker prager material model of plasticity and numerical integrating methods abstract internal symmetries for the drucker prager material model and numerical methods based on them are investigated here we first convert the non linear constitutive equations to a lie type system x a x t x where a sl 5 2 r is a lie algebra of the special orthochronous pseudo linear group sl 5 2 r the underlying space of drucker prager model in the plastic phase is a pseudo sphere in the pseudo riemann manifold which is locally a pseudo euclidean space e 7 5 2 and whose metric tensor is indefinite having signature 5 2 and also depends on the temporal component then we split the drucker prager yield condition into two sub yield conditions which together with two integrating factors idea led us to derive two lie type systems in the product space m 5 1 m 1 1 the lie algebra is the direct sum so 5 1 so 1 1 and the symmetry group is thus soo 5 1 soo 1 1 these results are essential from computational aspect based on these symmetry groups exponential mappings are developed which update stress points exactly on the yield surface at every time increment without any iteration as an example the results calculated by using the two different group preserving schemes are compared to that calculated by the runge kutta scheme together with the newton raphson iterative solution of a non linear algebraic equation which enforcing the consistency condition

non linear b stability and symmetry preserving return mapping algorithms for plasticity and viscoplasticity a class of second order accurate return mapping algorithms is presented which lead to symmetric algorithmic tangent moduli and contain the classical backward euler return maps as a particular case more importantly it is shown that this class of return maps is contractive relative to the natural norm defined by the complementary helmholz free energy function b stability since the equations of classical plasticity and viscoplasticity are shown to be contractive relative to this natural norm the requirement of b stability furnishes the appropriate notion of unconditionally stable algorithms for plasticity and viscoplasticity the analysis that follows depends critically on the assumption of convexity in particular the models of plasticity and viscoplasticity considered obey the principle of maximum plastic dissipation the proposed algorithms obey the discrete counterpart of this classical principle

a fast implementation of the feti dp method feti dp rbs lna and applications on large scale problems with localized non linearities as parallel and distributed computing gradually becomes the computing standard for large scale problems the domain decomposition method dd has received growing attention since it provides a natural basis for splitting a large problem into many small problems which can be submitted to individual computing nodes and processed in a parallel fashion this approach not only provides a method to solve large scale problems that are not solvable on a single computer by using direct sparse solvers but also gives a flexible solution to deal with large scale problems with localized non linearities when some parts of the structure are modified only the corresponding subdomains and the interface equation that connects all the subdomains need to be recomputed in this paper the dual primal finite element tearing and interconnecting method feti dp is carefully investigated and a reduced back substitution rbs algorithm is proposed to accelerate the time consuming preconditioned conjugate gradient pcg iterations involved in the interface problems linear non linear analysis lna is also adopted for large scale problems with localized non linearities based on subdomain linear non linear identification criteria this combined approach is named as the feti dp rbs lna algorithm and demonstrated on the mechanical analyses of a welding problem serial cpu costs of this algorithm are measured at each solution stage and compared with that from the ibm watson direct sparse solver and the feti dp method the results demonstrate the effectiveness of the proposed computational approach for simulating welding problems which is representative of a large class of three dimensional large scale problems with localized non linearities

a direct complementarity approach for the elastoplastic analysis of plane stress and plane strain structures this paper presents a direct complementarity approach for carrying out the elastoplastic analysis of plane stress and plane strain structures founded on a traditional finite step formulation our approach however avoids the typically cumbersome implementation of iterative predictorcorrector procedures associated with the ubiquitous return mapping algorithm instead at each predefined step the governing formulationcast in its most natural mathematical programming format known as a mixed complementarity problemis directly solved by using a complementarity solver run from within a mathematical modeling system we have chosen the industry standard general algebraic modeling system path mixed complementarity problem solver that is called from within the general algebraic modeling system environment we consider both von mises and tresca materials with perfect or hardening kinematic and isotropic behaviors our numerical tests five benchmark examples of which are presented in this paper have been carried out using models constructed from the mixed finite element of capsoni and corradi comput methods appl mech eng 1997 141 6793 which beneficially offers a locking free behavior and coarse mesh accuracy the results indicate in addition to an isochoric locking free behavior good accuracy and the ability to circumvent the difficult singularity problem associated with the corners of tresca yield surfaces

a component based hybrid reduced basis finite element method for solid mechanics with local nonlinearities abstract the scrbe static condensation reduced basis element method is a component to system model order reduction approach for efficient many query and real time treatment of linear partial differential equations characterized by many spatially distributed constitutive geometry and topology parameters in this paper we incorporate the scrbe approach into a framework for analysis of problems in solid mechanics which are largely linear with the exception of local nonlinearities in particular we exploit a linear nonlinear domain decomposition to develop a hybrid formulation we consider a scrbe approximation over the assumed predominant part of the domain associated to a linear elasticity model we revert to a full finite element fe approximation over the part of the domain associated to the locally nonlinear model we adapt the scrbe port training procedures to anticipate the behavior of the field over the linear nonlinear interface and hence ensure an accurate solution over the entire domain we choose a globally conforming approximation which exploits the intrinsic port structure of the scrbe method and yields a decoupled non invasive formulation of the respective linear and nonlinear blocks of the residual vector and newton jacobian matrix we present numerical results for several local nonlinearities elastic contact with and without friction plastic contact which demonstrate substantial savings relative to standard fe approximation over the entire domain in particular in the regime in which the linear part of the domain represents the majority of the fe degrees of freedom

a class of mixed assumed strain methods and the method of incompatible modes a three field mixed formulation in terms of displacements stresses and an enhanced strain field is presented which encompasses as a particular case the classical method of incompatible modes within this frame work incompatible elements arise as particular compatible mixed approximations of the enhanced strain field the conditions that the stress interpolation contain piece wise constant functions and be l2 ortho gonal to the enhanced strain interpolation ensure satisfaction of the patch test and allow the elimination of the stress field from the formulation the preceding conditions are formulated in a form particularly convenient for element design as an illustration of the methodology three new elements are developed and shown to exhibit good performance a plane 3d elastic plastic quad an axisymmetric element and a thick plate bending quad the formulation described herein is suitable for non linear analysis

second order accurate integration algorithms for von mises plasticity with a nonlinear kinematic hardening mechanism abstract two second order numerical schemes for von mises plasticity with a combination of linear isotropic and nonlinear kinematic hardening are presented the first scheme is the generalized midpoint integration procedure originally introduced by ortiz and popov in 1985 detailed and applied here to the case of armstrong frederick nonlinear kinematic hardening the second algorithm is based on the constitutive model exponential based reformulation and on the integration procedure previously introduced by the authors in the context of linearly hardening materials there are two main targets to the work firstly we wish to extensively test the generalized midpoint procedure since in the scientific literature no thorough numerical testing campaign has been carried out on this second order algorithm secondly we wish to extend the exponential based integration technique also to nonlinear hardening materials a wide numerical investigation is carried out in order to compare the performance of the two methods

on the generalized q markov cover models for discrete time systems a new method for obtaining generalized q markov cover models for discrete time siso systems is proposed and is based on the inverse solution of the lyapunov equation the reduced order models not only match the markov parameters and high frequency power moments like the conventional q markov cover techniques but also match time moments and low frequency power moments of the original system