equivalent spatial and material descriptions of finite deformation elastoplasticity in principal axes abstract a spatial description of the theory of rate independent finite deformation elastoplasticity in which the stress tensor is defined through the strain energy function is discussed the main assumption of isotropic elastic response and invariance requirements under superposed rigid body motion restrict the acceptable forms of the strain energy function to those given in terms of principal values of the strain measure of elastic distortion the formulation is developed on a manifold and the corresponding material description is obtained simply by pull back of the derived spatial form by appealing to the notion of covariance the method of principal axes is systematically exploited to derive the explicit expression for the stress tensor computation for an arbitrary form of the strain energy function and the explicit form of the evolution equation for an arbitrary form of the yield function a model problem of volume preserving plastic flow is discussed in the closure

numerical implementation of multiplicative elasto plasticity into assumed strain elements with application to shells at large strains alternative formulations of isotropic large strain elasto plasticity are presented which are especially well suited for the implementation into assumed strain elements based on the multiplicative decomposition of the deformation gradient into elastic and plastic parts three distinct eigenvalue problems related to the reference intermediate and current configuration are investigated these eigenvalue problems are connected by similarity transformations which preserve the eigenvalues they play an important role in the subsequent development of alternative constitutive formulations and the corresponding finite element implementation the developed constitutive procedures rely on the right cauchy green tensor or equivalently on the green lagrangian strain tensor rather than the deformation gradient consequently they can be applied directly to assumed strain elements specifically we are concerned with efficient low order shell elements for which the assumed strain method has proven to be extremely powerful to overcome spurious locking effects

multi scale second order computational homogenization of multi phase materials a nested finite element solution strategy this paper presents the detailed implementation and computational aspects of a novel second order computational homogenization procedure which is suitable for a multi scale modelling of macroscopic localization and size effects the second order scheme is an extension of the classical first order computational homogenization framework and is based on a proper incorporation of the gradient of the macroscopic deformation gradient tensor into the kinematical macro micro scale transition from the microstructural analysis the macroscopic stress and higher order stress tensors are obtained thus delivering a microstructurally based constitutive response of the macroscopic second gradient continuum the higher order macroscopic constitutive tangents are derived through static condensation of the microscopic global tangent matrix for the solution of the second gradient equilibrium problem on the macrolevel a mixed finite element formulation is developed as an example the second order computational homogenization approach is applied for the multi scale analysis of simple shear of a constrained heterogeneous strip where a pronounced boundary size effect appears

on free energy based formulations for kinematic hardening and the decomposition f fpfe within the framework of linear plasticity based on additive decomposition of the linear strain tensor kinematical hardening can be described by means of extended potentials the method is elegant and avoids the need for evolution equations the extension of small strain formulations to the finite strain case which is based on the multiplicative decomposition of the deformation gradient into elastic and inelastic parts proved not straight forward specifically the symmetry of the resulting back stress remained elusive in this paper a free energy based formulation incorporating the effect of kinematic hardening is proposed the formulation is able to reproduce symmetric expressions for the back stress while incorporating the multiplicative decomposition of the deformation gradient kinematic hardening is combined with isotropic hardening where an associative flow rule and von mises yield criterion are applied it is shown that the symmetry of the back stress is strongly related to its treatment as a truly spatial tensor where contraction operations are to be conducted using the current metric the latter depends naturally on the deformation gradient itself various numerical examples are presented

on the finite element method for singularly perturbed reaction diffusion problems in two and one dimensions second order elliptic boundary value problems which are allowed to degenerate into zero order equations are considered the behavior of the ordinary galerkin finite element method without special arrangements to treat singularities is studied as the problem ranges from true second order to singularly perturbed

on deformational and configurational mechanics of micromorphic hyperelasticity theory and computation abstract a micromorphic continuum formulation is presented in the context of both the spatial and the material motion problem for both approaches the kinematics as well as the balance relations together with the various representations of the occurring stress fields are derived the relations between the spatial motion problem and the material motion problem quantities are examined in detail upon a hyperelastic constitutive assumption a finite element approximation is derived and the material force method which is especially suited for defect mechanics problems is successfully applied to the present micromorphic continuum theory

a uniformly convergent method for a singularly perturbed semilinear reaction diffusion problem with multiple solutions this paper considers a simple central difference scheme for a singularly perturbed semilinear reaction diffusion problem which may have multiple solutions asymptotic properties of solutions to this problem are discussed and analyzed to compute accurate approximations to these solutions we consider a piecewise equidistant mesh of shishkin type which contains o n points on such a mesh we prove existence of a solution to the discretization and show that it is accurate of order n 2 in 2 n in the discrete maximum norm where the constant factor in this error estimate is independent of the perturbation parameter e and n numerical results are presented that verify this rate of convergence

finite elements for materials with strain gradient effects a finite element implementation is reported of the fleck hutchinson phenomenological strain gradient theory this theory fits within the toupin mindlin framework and deals with first order strain gradients and the associated work conjugate higher order stresses in conventional displacement based approaches the interpolation of displacement requires c1 continuity in order to ensure convergence of the finite element procedure for higher order theories mixed type finite elements are developed herein for the fleck hutchinson theory these elements use standard c0 continuous shape functions and can achieve the same convergence as c1 elements these c0 elements use displacements and displacement gradients as nodal degrees of freedom kinematic constraints between displacement gradients are enforced via the lagrange multiplier method the elements developed all pass a patch test the resulting finite element scheme is used to solve some representative linear elastic boundary value problems and the comparative accuracy of various types of element is evaluated copyright 1999 john wiley sons ltd

a shell finite element for large strain elastoplasticity with anisotropies part i shell theory and variational principle a shell model for finite elastic and finite plastic strains is derived taking into account initial and induced anisotropies a corresponding eight node c 0 shell element with three displacement and three director degrees of freedom at each node is developed which combines the advantages of an isoparametric description of geometry and deformation with an effective plane stress formulation the element accounts for isochoric or approximately isochoric deformation due to finite plastic strains because of the three displacement and three director degrees of freedom at each node it is easily possible to link different parts of a composed irregular shell structure or to connect the derived shell element with solid brick elements this paper presents the shell theory based on the kinematics of finite elastoplasticity proposed in schieck and stumpf 1995 and the special geometric concept of the derived shell model the lagrange multiplier method is applied to introduce into the virtual work principle the transverse normality constraint and the condition of isochoric deformation where the lagrange multipliers can be condensed inside the element procedure various assumed strain techniques designed to avoid the membrane locking are compared with known methods in the literature according to the numerical experience so far the proposed shell finite element is free of locking effects and spurious modes part ii presents the constitutive equations for finite elastic plastic strains accounting for initial and induced anisotropies and the implementation of the model into the fe code a comprehensive set of numerical examples is provided involving the tension of a plane specimen with necking and shear band localization the elastic plastic response of a simply supported plate with a localization of the plastic bending strains in the four corner zones the elastic plastic deformation mode of the so called scordelis lo roof and the elastic plastic buckling of a cylindrical shell showing an essential influence of the anisotropic material behavior the results illustrate the performance of the proposed shell finite element for a wide range of engineering applications

approximate controllability of semi linear neutral integro differential systems with finite delay abstract this paper considers the approximate controllability of semilinear neutral integrodifferential systems with finite delay in hilbert space since the considered equation admits nonlinear terms involving spatial derivatives the fraction power theory and norm are used to discuss the problem so that the established results can be applied to them the results are obtained by using the sadovskii fixed point theorem and the theory of analytic resolvent operator an example is presented to illustrate the applications of the obtained results

uniform error estimates in the finite element method for a singularly perturbed reaction diffusion problem consider the problem e 2 u u f with homogeneous neumann boundary condition in a bounded smooth domain in r n the whole range 0 e 1 is treated the galerkin finite element method is used on a globally quasi uniform mesh of size h the mesh is fixed and independent of e a precise analysis of how the error at each point depends on h and e is is presented as an application first order error estimates in h which are uniform with respect to e are given

a formulation of finite elastoplasticity based on dual co and contra variant eigenvector triads normalized with respect to a plastic metric the article presents a new formulation of isotropic elastoplasticity at large strains in both the lagrangian and the eulerian geometric setting and addresses aspects of its numerical implementation the key ingredients on the theoretical side are the introduction of a plastic metric for the description of the local history dependent inelastic material response the definition of a convex elastic domain in the space of the local stress like variable conjugate to the plastic metric denoted as the plastic force and a fully equivalent lagrangian and eulerian representation of all constitutive functions in spectral form for general non cartesian coordinate charts in terms of dual co and contra variant eigenvector triads which are normalized with respect to the plastic metric on the numerical side we propose a stress update algorithm for general non associative isotropic elasto visco plastic response with an arbitrary number of scalar internal variables the algorithm is based on an exponential map integrator and is recast into a general return mapping scheme methodically organized with tensorial pre and postprocessing and a constitutive box in the eigenvalue space furthermore we propose a new perturbation stabilization technique which dramatically enhances the convergence of the general return algorithm the theoretical and numerical developments are applied to a constitutive model problem with large elastic and large plastic strains the von mises tresca type associative plastic flow in ogden type large strain elastic materials

application of material forces to hyperelastostatic fracture mechanics i continuum mechanical setting the concern of this work is a consequent exploitation of the notion of material forces for the application within hyperelastostatic fracture mechanics contrary to physical forces material forces act on the material manifold thus essentially representing the tendency of defects like cracks or inclusions to move relative to the ambient material based on the formulation of the appropriate quasi static balance laws in the material space we aim at a fresh look onto classical aspects of hyperelastostatic fracture mechanics operating throughout within the geometrically nonlinear setting we emphasize on the one hand the duality of the direct and the inverse motion description and on the other hand we re establish the classical path integrals from elementary equilibrium considerations in the material space

existence of solutions for a class of neutral partial differential equations with nonlocal conditions in the α norm abstract this paper is concerned with the existence of mild and strong solutions for some neutral partial differential equations with nonlocal conditions in the norm the linear part of the equations is assumed to generate a compact analytic semigroup of uniformly bounded linear operators and the nonlinear part satisfies some lipschitz conditions with respect to the norm by using a fixed point theorem of sadovskii we establish some existence results which generalize the recent conclusions on this issue in the end the obtained results will be applied to a class of partial differential equations with nonlocal conditions

on theoretical and numerical methods in the theory of porous media based on polar and non polar elasto plastic solid materials the consideration of saturated and non saturated porous solid materials as for instance soil concrete sinter materials polymeric and metallic foams temperate ice living tissues etc naturally falls into the category of multiphase materials which can be described by use of a macroscopic continuum mechanical approach within the well founded framework of the theory of porous media tpm in the present contribution granular elasto plastic porous solid skeletons frictional materials are taken into consideration where regardless of whether or not the solid is fluid saturated or empty localization phenomena can occur as a result of local concentrations of plastic strains as a consequence of localization phenomena the numerical solution of the governing equations generally reveals an ill posed problem in particular the shear band width strongly depends on the mesh size of the finite element discretization by the fact that each mesh refinement leads to a decrease of the shear band width until one obtains a singular surface in the present article it is shown that the inclusion of fluid viscosity in the saturated case and the inclusion of micropolar grain rotations both in the saturated and in the non saturated case leads to a regularization of the shear band problem on the other hand the inclusion of micropolar degrees of freedom in the sense of the cosserat brothers additionally allows for the determination of the local average grain rotations the numerical examples are solved by use of finite element discretization techniques where in particular the computation of shear band localization phenomena is carried out by the example of the well known geotechnical slope failure problem and two additional academic problems which clearly demonstrate the efficiency of the proposed procedure

on a path following method for non linear solid mechanics with applications to structural and cardiac mechanics subject to arbitrary loading scenarios abstract in computational solid mechanics path following methods have been proven useful when dealing with non monotonously evolving loading magnitudes as encountered e g in instability or softening behaviour the present work derives from a displacement based method a cavity volume based path following method and considers its application specifically to cardiac mechanics problems both methods are able to account for an arbitrary number of simultaneously acting loading conditions when applied to the newton raphson method the corresponding loading increments are computed by means of volume increments or point wise prescribed displacement increments respectively no additional variables are required and the physics of the problem at hands is not altered both methods are implemented in an in house meshfree modelling software and successfully applied to non linear elastic and inelastic problems in structural mechanics the cavity volume control method in particular is demonstrated to accurately predict the highly non linear elastic and anisotropic material behaviour encountered when modelling the heart albeit the proposed methods can be equally used e g in finite element methods they are very well suited for meshfree methods where the kronecker delta property does not apply

multiscale computational homogenization methods with a gradient enhanced scheme based on the discontinuous galerkin formulation abstract when considering problems of dimensions close to the characteristic length of the material the size effects can not be neglected and the classical so called first order multiscale computational homogenization scheme fmch looses accuracy motivating the use of a second order multiscale computational homogenization smch scheme this second order scheme uses the classical continuum at the micro scale while considering a second order continuum at the macro scale although the theoretical background of the second order continuum is increasing the implementation into a finite element code is not straightforward because of the lack of high order continuity of the shape functions in this work we propose a smch scheme relying on the discontinuous galerkin dg method at the macro scale which simplifies the implementation of the method indeed the dg method is a generalization of weak formulations allowing for inter element discontinuities either at the c 0 level or at the c 1 level and it can thus be used to constrain weakly the c 1 continuity at the macro scale the c 0 continuity can be either weakly constrained by using the dg method or strongly constrained by using usual c 0 displacement based finite elements therefore two formulations can be used at the macro scale i the full discontinuous galerkin formulation fdg with weak c 0 and c 1 continuity enforcements and ii the enriched discontinuous galerkin formulation edg with high order term enrichment into the conventional c 0 finite element framework the micro problem is formulated in terms of standard equilibrium and periodic boundary conditions a parallel implementation in three dimensions for non linear finite deformation problems is developed showing that the proposed method can be integrated into conventional finite element codes in a straightforward and efficient way

a shell finite element for large strain elastoplasticity with anisotropies part ii constitutive equations and numerical applications abstract an eight node c 0 shell element for finite elastic plastic deformations with anisotropies is developed it combines the advantages of an isoparametric description of geometry and deformation the application of tensors in cartesian components and a real and effective plane stress description with three displacement and three director degrees of freedom at each node in part i of the paper the shell theory including the kinematics the variational principle the application of lagrange multipliers with their condensation on the element level and a comparative study of various assumed strain techniques were presented and the results of convergence tests given in this paper we consider the constitutive equations for large elastic and large plastic strains accounting for initial and induced anisotropies and the corresponding thermodynamics then we investigate the return algorithm for finite strains and the implementation of the element procedure including stiffness matrix and residual force vector finally we present the results of extended numerical applications and a comparison with fe solutions published in the literature as far as such are available

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

micropolar hyper elastoplasticity constitutive model consistent linearization and simulation of 3d scale effects summary a computational model for micropolar hyperelastic based finite elastoplasticity that incorporates isotropic hardening is developed the basic concepts of the non linear micropolar kinematic framework are reviewed and a thermodynamically consistent constitutive model that features neo hooke type elasticity and generalized von mises plasticity is described the integration of the constitutive initial value problem is carried out by means of an elastic predictor plastic corrector algorithm which retains plastic incompressibility the solution procedure is developed carefully and described in detail the consistent material tangent is derived the micropolar constitutive model is implemented in an implicit finite element framework the numerical example of a notched cylindrical bar subjected to large axial displacements and large twist angles is presented the results of the finite element simulations demonstrate i that the methodology is capable of capturing the size effect in three dimensional elastoplastic solids in the finite strain regime ii that the formulation possesses a regularizing effect in the presence of strain localization and iii that asymptotically quadratic convergence rates of the newton raphson procedure are achieved throughout this paper effort is made to present the developments as a direct extension of standard finite deformation computational plasticity copyright 2012 john wiley sons ltd

a hyperelastic based large strain elasto plastic constitutive formulation with combined isotropic kinematic hardening using the logarithmic stress and strain measures summary this paper addresses the formulation of a set of constitutive equations for finite deformation metal plasticity the combined isotropic kinematic hardening model of the infinitesimal theory of plasticity is extended to the large strain range on the basis of three main assumptions i the formulation is hyperelastic based ii the stress strain law preserves the elastic constants of the infinitesimal theory but is written in terms of the hencky strain tensor and its elastic work conjugate stress tensor and iii the multiplicative decomposition of the deformation gradient is adopted since no stress rates are present the formulation is of course numerically objective in the time integration it is shown that the model gives adequate physical behaviour and comparison is made with an equivalent constitutive model based on the additive decomposition of the strain tensor

generalized continuum modeling of 2 d periodic cellular solids abstract a generalized continuum representation of two dimensional periodic cellular solids is obtained by treating these materials as micropolar continua linear elastic micropolar constants are obtained using an energy approach for square equilateral triangular mixed triangle and diamond cell topologies the constants are obtained by equating two different continuous approximations of the strain energy function furthermore the effects of shear deformation of the cell walls on the micropolar elastic constants are also discussed a continuum micropolar finite element approach is developed for numerical simulations of the cell structures the solutions from the continuum representation are compared with the exact discrete simulations of these cell structures for a model problem of elastic indentation of a rectangular domain by a point force the utility of the micropolar continuum representation is illustrated by comparing various cell structures with respect to the stress concentration factor at the root of a circular notch

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

pointwise approximation of corner singularities for a singularly perturbed reaction diffusion equation in an l shaped domain a singularly perturbed reaction diffusion equation is posed in a two dimensional l shaped domain subject to a continuous dirchlet boundary condition its solutions are in the holder space c 2 3 and typically exhibit boundary layers and corner singularities the problem is discretized on a tensor product shishkin mesh that is further refined in a neighboorhood of the vertex of angle 3 2 we establish almost second order convergence of our numerical method in the discrete maximum norm uniformly in the small diffusion parameter numerical results are presented that support our theoretical error estimate

existence and regularity of solutions for some neutral partial differential equations with nonlocal conditions abstract in this work we establish several results about the existence and regularity of solutions for some neutral partial differential equations with nonlocal conditions we assume that the linear part generates a strongly continuous semigroup on a general banach space

a model of finite strain viscoplasticity based on unified constitutive equations theoretical and computational considerations with applications to shells abstract a model of multiplicative finite strain viscoplasticity is developed hereby the elastic internal potential is assumed to depend quadratically on the elastic strain measure ce by means of corresponding invariants hence possible extension of the model to take anisotropy into account is straightforward the evolution equation of the unified type due to bodner and partom asme j appl mech 42 1975 385 are modified so as to fit into the framework adopted the numerical treatment of the problem is fully developed specifically the algorithmic aspects of the resulting non additive structure are intensively discussed hereby the treatment of the exponential map of a non symmetric argument is of a central importance two consistent methods are developed various applications to shell problems are considered hereby a shell theory with seven degrees of freedom together with a 4 node enhanced strain finite element formulation are used a central feature of the shell formulation is its eligibility to the application of a three dimensional constitutive law

a computational model for elasto viscoplastic solids at finite strain with reference to thin shell applications this work extends a previously developed methodology for computational plasticity at finite strains that is based on the exponential map and logarithmic stretches to the context of isotropic elasto viscoplastic solids a particular form of the strain energy function given in terms of its principal values is employed it is noticeable that within the proposed framework the small strain integration algorithms and the corresponding consistent tangent operators automatically extend to the finite strain regime central to the effort of this formulation is the derivation of the closed form of a tangent modulus obtained by linearization of incremental non linear problem this ensures asymptotically quadratic rates of convergence of the newton raphson procedure in the implicit finite element solution to illustrate the performance of the presented formulation several numerical examples involving failure by strain localization and finite deformations are given 1998 john wiley sons ltd

application of material forces to hyperelastostatic fracture mechanics ii computational setting the concern of this work is a novel algorithmic treatment of hyperelastostatic fracture mechanics problems consistent to the notion of material forces within the geometrically nonlinear setting of continuum mechanics to this end we consider the continuum mechanics of material forces as outlined in part i of this work p steinmann int j solid struct 37 7371 7391 which act contrary to the common physical forces on the material manifold or rather in the material space in the sequel it is proposed to discretize the corresponding quasi static balance of pseudo momentum by a standard galerkin finite element procedure as a result we obtain global discrete node point quantities the material node point forces which prove to be of the same qualitative and quantitative importance for the assessment of fracture mechanics problems as the classical j integral

an ale formulation based on spatial and material settings of continuum mechanics part 1 generic hyperelastic formulation abstract the present contribution deals with the derivation of a generic hyperelastic arbitrary lagrangian eulerian formulation on the basis of a consistent variational framework the governing equations follow straightforwardly from the dirichlet principle for conservative mechanical systems thereby the key idea is the reformulation of the total variation of the potential energy at fixed referential coordinates in terms of its variation at fixed material and at fixed spatial coordinates the corresponding euler lagrange equations define the spatial and the material motion version of the balance of linear momentum i e the balance of spatial and material forces in a consistent dual format in the discretised setting the governing equations are solved simultaneously rendering the spatial and the material configuration which minimise the overall potential energy of the system the remeshing strategy of the ale formulation is thus no longer user defined but objective in the sense of energy minimisation if the governing equations are derived from a potential i e either from an incremental potential or from a total potential as in the present case they are inherently symmetric both in the continuous case and in the discrete case this symmetry property is particularly appealing since it ensures symmetric system matrices upon discretisation

on anisotropic flow rules in multiplicative elastoplasticity at finite strains abstract hill s anisotropic formulation of the flow rule is extended so as to fit in the realm of multiplicative finite strain plasticity the anisotropic hill type yield criterion is formulated in terms of purely material quantities the list of arguments of the flow function includes a non symmetric material eshelby like stress tensor as well as structural tensors that describe the anisotropy at hand the formulation is exemplified on orthotropy where three structural tensors are employed the fact that the stress tensor is not symmetric necessitates a special treatment of the flow function where representation theorems of tensor valued function with non symmetric arguments are invoked the consequences of such a definition on the resulting inelastic rate are discussed in full it is shown that the corresponding resulting rate as defined at the actual configuration is not symmetric any more accordingly the rate naturally includes a plastic material spin moreover we deal with the theoretically interesting question of how to define spin free rates it is also demonstrated that the flow function must depend not only on the stress tensor and on adequate structural tensors but also on the deformation itself in form of the right cauchy green tensor c however this surprising dependency which must obey a specific form can be justified as physically meaningful various numerical examples of large plastic deformations of structural components are presented that underpin the capabilities of the formulation

a continuum based shell theory for non linear applications the paper introduces a non linear shell theory which provides a complete three dimensional state of stress since the theory is derived from simple three dimensional continuum mechanics it is very easy to understand as an example the theory is applied to quadrilateral shell elements which provide only displacement degrees of freedom located at nodes on the outer surfaces and one degree of freedom at the middle surface it is proposed to eliminate this degree of freedom on element level so that the elements have the same layout as the equivalent brick elements but have a better behaviour in bending have stress resultants and are cheaper with respect to computational effort the advantages with respect to implementation in a finite element program as well as in special applications are obvious however well known conditioning problems in thin shell applications must be expected therefore emphasis is put on this issue in the example problems it is shown that the elements can give acceptable answers in engineering applications and offer a potential for material non linear applications which will be considered in a forthcoming paper

on the mathematical foundation of the 1 1 2 platemodel abstract the plate problem of three dimensional 3 d linearized elastostatics is considered inthe framework of the hierarchical modelling with the help of the energy projection method it isshown that the 1 1 2 bending model is the simplest asymptotically correct model in thehierarchical family i e the distance between the solution of the 3 d problem and themodel solution in the energy norm tends to zero for vanishing plate thickness h whileusing unmodified 3 d material laws the formulation of the 1 1 2 bending model and the mechanical significance of itsingredients are discussed we present error estimates for the deviation of the 1 1 2 solutionfrom the kirchhoff solution as well as from the 3 d solution by using the 1 1 2 energy normand the 3 d energy norm respectively the analysis leads to h as order of convergence the results are illustrated by a numerical example

formulation of implicit finite element methods for multiplicative finite deformation plasticity some constitutive and computational aspects of finite deformation plasticity are discussed attention is restricted to multiplicative theories of plasticity in which the deformation gradients are assumed to be decomposable into elastic and plastic terms it is shown by way of consistent linearization of momentum balance that geometric terms arise which are associated with the motion of the intermediate configuration and which in general render the tangent operator non symmetric even for associated plastic flow both explicit i e no equilibrium iteration and implicit finite element formulations are considered an assumed strain formulation is used to accommodate the near incompressibility associated with fully developed isochoric plastic flow as an example of explicit integration the rate tangent modulus method is reviewed in some detail an implicit scheme is derived for which the consistent tangents resulting in quadratic convergence of the equilibrium iterations can be written out in closed form for arbitrary material models all the geometrical terms associated with the motion of the intermediate configuration and the treatment of incompressibility are given explicitly examples of application to void growth and coalescence and to crack tip blunting are developed which illustrate the performance of the implicit method

a parameter robust numerical method for a two dimensional reaction diffusion problem in this paper a singularly perturbed reaction diffusion partial differential equation in two space dimensions is examined by means of an appropriate decomposition we describe the asymptotic behaviour of the solution of problems of this kind a central finite difference scheme is constructed for this problem which involves an appropriate shishkin mesh we prove that the numerical approximations are almost second order uniformly convergent in the maximum norm with respect to the singular perturbation parameter some numerical experiments are given that illustrate in practice the theoretical order of convergence established for the numerical method

a second order homogenization procedure for multi scale analysis based on micropolar kinematics the paper presents a higher order homogenization scheme based on non linear micropolar kinematics representing the macroscopic variation within a representative volume element rve of the material on the microstructural level the micro macro kinematical coupling is introduced as a second order taylor series expansion of the macro displacement field and the microstructural displacement variation is gathered in a fluctuation term this approach relates strongly to second gradient continuum formulations presented by e g kouznetsova et al int j numer meth engng 2002 54 1235 1260 thus establishing a link between second gradient and micropolar theories the major difference of the present approach as compared to second gradient formulations is that an additional constraint is placed on the higher order deformation gradient in terms of the micropolar stretch the driving vehicle for the derivation of the homogenized macroscopic stress measures is the hill mandel condition postulating the equivalence of microscopic and macroscopic homogenized virtual work thereby the resulting homogenization procedure yields not only a stress tensor conjugated to the micropolar stretch tensor but also the couple stress tensor conjugated to the micropolar curvature tensor the paper is concluded by a couple of numerical examples demonstrating the size effects imposed by the homogenization of stresses based on the micropolar kinematics copyright 2006 john wiley sons ltd

on the numerical implications of multiplicative inelasticity with an anisotropic elastic constitutive law the statement that theories of inelasticity at finite strains have arrived at a high level of development is only true in conjunction with isotropic material behaviour from both points of view theoretical and computational the extension to anisotropic material behaviour seems to be a complicated task the statement is especially true when the multiplicative decomposition of the deformation gradient is considered a basis for the formulation of special interest are questions related to the mathematical form of the stored energy function or equivalently of the constitutive relation for the material stress tensor as the thermodynamical force this paper deals with the above issues the anisotropic formulation is accomplished using the notion of structural tensors here we suggest that the privileged directions of the material should be transformed in a specific way under the action of the inelastic part of the deformation gradient the inelastic behaviour is assumed to be governed by evolution equations of the unified type numerically we deal with the full multiplicative structure of the theory the numerical treatment is developed in full detail expressions concerning the local iteration as well as the tangent operator are derived various numerical examples with applications to shells are presented which demonstrate the influence of anisotropy and the applicability of the theory

a micropolar theory of finite deformation and finite rotation multiplicative elastoplasticity the aim of this work is to formulate a geometrically exact theory of finite deformation and finite rotation micropolar elastoplasticity to obtain a generalized nonlinear continuum framework to this end the classical deformation map is supplemented by an independent rotation field to yield an enhanced configuration space thereby the rotational part of the formulation is consequently parameterized in terms of the rotation pseudo vector via the euler rodrigues formula then micropolar hyperelasticity and multiplicative elastoplasticity are conceptionally derived as in the classical boltzmann continuum the proposed theory is consequently developed in a modern geometry oriented fashion linearization of the kinematics retrofits the well known structure of the micropolar geometrically linear theory

maximum norm error analysis of a 2d singularly perturbed semilinear reaction diffusion problem a semilinear reaction diffusion equation with multiple solutions is considered in a smooth two dimensional domain its diffusion parameter e 2 is arbitrarily small which induces boundary layers constructing discrete sub and super solutions we prove existence and investigate the accuracy of multiple discrete solutions on layer adapted meshes of bakhvalov and shishkin types it is shown that one gets second order convergence with in the case of the shishkin mesh a logarithmic factor in the discrete maximum norm uniformly in e for 0 is the maximum side length of mesh elements while the number of mesh nodes does not exceed ch 2 numerical experiments are performed to support the theoretical results

implicit finite element formulation of multiresolution continuum theory abstract the multiresolution continuum theory is a higher order continuum theory where additional kinematic variables account for microstructural inhomogeneities at several distinct length scales this can be particularly important for localization problems the strength of this theory is that it can account for details in the microstructure of a material without using an extremely fine mesh the present paper describes the implementation and verification of a 3d elastic plastic multiresolution element based on an implicit time stepping algorithm it is implemented in the general purpose finite element program feap the mesh independency associated with the length scale parameter is examined and the convergence rate of the element is also evaluated

stabilised approximation of interior layer solutions of a singularly perturbed semilinear reaction diffusion problem a semilinear reaction diffusion two point boundary value problem whose second order derivative is multiplied by a small positive parameter varepsilon 2 is considered it can have multiple solutions the numerical computation of solutions having interior transition layers is analysed it is demonstrated that the accurate computation of such solutions is exceptionally difficult to address this difficulty we propose an artificial diffusion stabilization for both standard and stabilised finite difference methods on suitable shishkin meshes we prove existence and investigate the accuracy of computed solutions by constructing discrete sub and super solutions convergence results are deduced that depend on the relative sizes of varepsilon and n where n is the number of mesh intervals numerical experiments are given in support of these theoretical results practical issues in using newton s method to compute a discrete solution are discussed

finite element approximation of convection diffusion problems using an exponentially graded mesh we present the analysis of an h version finite element method for the approximation of the solution to convection diffusion problems the method uses piece wise polynomials of degree p 1 defined on an exponentially graded mesh optimally constructed for the approximation of exponential layers we consider a model convection diffusion problem posed on the unit square and establish robust optimal convergence rates in the energy and in the maximum norm we also present the results of some numerical computations that illustrate our theoretical findings and compare the proposed method with others found in the literature

on theory and numerics of large viscoplastic deformation abstract a multiplicative theory and corresponding computations of finite elastic viscoplastic deformations based on unified constitutive equations are presented basic features of theory are 1 the inelastic part of the deformation gradient is understood as a material stretch type tensor no need arises for the notion of the intermediate configuration 2 in the isotropic case the use of an elastic logarithmic strain tensor is shown to lead to the identification that the elastic strains are given by the quantity c 1 p c with c p being an inelastic cauchy green type tensor 3 in spite of the fact that the logarithmic strain measure is used closed forms of the tangent operator within an implicit time integration scheme are given circumventing very involved elaborations when the spectral decomposition is used 4 evolution equations of the bodner and partom type are employed a general formalism is developed for the application of the evolution laws of the unified type within the theoretical framework the formalism can be used to generalize any unified evolution equations formulated for infinetismal strains to the range of finite strains 5 for the axisymmetric case in addition to the displacement based 4 node and 9 node finite elements an enhanced strain 4 node element is presented it is shown anyhow that in the highly nonlinear regime the enhanced strain formulation may exhibit numerical instabilities different numerical examples of finite strain deformations are presented

a second order overlapping schwarz method for a 2d singularly perturbed semilinear reaction diffusion problem an overlapping schwarz domain decomposition is applied to a semilinear reaction diffusion equation posed in a smooth two dimensional domain the problem may exhibit multiple solutions its diffusion parameter e2 is arbitrarily small which induces boundary layers the schwarz method invokes a boundary layer subdomain and an interior subdomain the narrow subdomain overlap being of width o e ln h where h is the maximum side length of mesh elements and the global number of mesh nodes does not exceed o h 2 we employ finite differences on layer adapted meshes of bakhvalov and shishkin types in the boundary layer subdomain and lumped mass linear finite elements on a quasiuniform delaunay triangulation in the interior subdomain for this iterative method we present maximum norm error estimates for e 0 1 it is shown in particular that when e c ln h 1 one iteration is sufficient to get second order convergence with in the case of the shishkin mesh a logarithmic factor in the maximum norm uniformly in e numerical results are presented to support our theoretical conclusions

nonlinear microstrain theories a hierarchy of higher order continua is presented that introduces additional degrees of freedom accounting for volume changes rotation and straining of an underlying microstructure an increase in the number of degrees of freedom represents a refinement of the material description in addition to available nonlinear cosserat and micromorphic theories general formulations of elastoviscoplastic behaviour are proposed for microdilatation and microstretch continua a microstrain theory is introduced that is based on six additional degrees of freedom describing the pure straining of the microstructural element in each case balance equations and boundary conditions are derived decompositions of the finite strain measures into elastic and plastic parts are provided the formulation of finite deformation elastoviscoplastic constitutive equations relies on the introduction of the free energy and dissipation potentials thus complying with requirements of continuum thermodynamics some guidelines for the selection of a suitable higher order model for a given material close the discussion

a brief survey on numerical methods for solving singularly perturbed problems in the present paper a brief survey on computational techniques for the different classes of singularly perturbed problems is given this survey is a continuation of work performed earlier by the first author and contains the literature of the work done by the researchers during the years 2000 2009 however some older important relevant papers are also included in this survey we also mentioned those papers which are not surveyed in the previous survey papers by the first author of this paper see appl math comput 30 1989 223 259 130 2002 457 510 134 2003 371 429 for details thus this survey paper contains a surprisingly large amount of literature on singularly perturbed problems and indeed can serve as an introduction to some of the ideas and methods for the singular perturbation problems

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

a parameter uniform numerical method for a singularly perturbed convection diffusion problem posed on an annulus abstract a finite difference method is constructed for a singularly perturbed convection diffusion problem posed on an annulus the method involves combining polar coordinates an upwind finite difference operator and a piecewise uniform shishkin mesh in the radial direction compatibility constraints are imposed on the data in the vicinity of certain characteristic points to ensure that interior layers do not form within the annulus a theoretical parameter uniform error bound is established and numerical results are presented to illustrate the performance of the numerical method applied to particular test problems

on a numerical implementation of a formulation of anisotropic continuum elastoplasticity at finite strains in a recent theoretical study see c sansour i karsaj j soric a formulation of anisotropic continuum elastoplasticity at finite strains part i modelling international journal of plasticity 22 2006 2346 2365 a constitutive model for anisotropic elastoplasticity at finite strains has been developed the model is based on the multiplicative decomposition of the deformation gradient the stored energy function as well as the flow rule has been considered as quadratic functions of their arguments in both cases the list of arguments is extended to include structural tensors which describe the anisotropy of the material response at hand non linear isotropic hardening is considered as well in this paper the integration of the constitutive law is presented the associative flow rule is integrated using the exponential map which preserves the plastic incompressibility condition the numerical treatment of the problem is fully developed and expressions related to the local iteration and the consistent tangent operator are considered in detail it is shown that while the consistent linearisation of the model is quite complicated it still can be achieved if various intriguing implicit dependencies are identified and correctly dealt with various numerical examples of three dimensional deformations of whole structural components are presented the examples clearly illustrate the influence of anisotropy on finite elastoplastic deformations

existence of solutions for neutral functional differential evolution equations with nonlocal conditions abstract in this paper by using fractional power of operators and sadovskii s fixed point theorem we study the existence of mild and strong solutions of semilinear neutral functional differential evolution equations with nonlocal conditions the results we obtained are a generalization and continuation of the recent results on this issue in the end an example is given to show the application of our results

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

existence of solutions for a class of abstract differential equations with nonlocal conditions abstract we discuss the existence of mild classical and strict solutions for a class of abstract differential equations with nonlocal conditions our technical approach allows the study of partial differential equations with nonlocal conditions involving partial derivatives or nonlinear expressions of the solution some concrete applications to partial differential equations are considered

multi scale constitutive modelling of heterogeneous materials with a gradient enhanced computational homogenization scheme a gradient enhanced computational homogenization procedure that allows for the modelling of microstructural size effects is proposed within a general non linear framework in this approach the macroscopic deformation gradient tensor and its gradient are imposed on a microstructural representative volume element rve this enables us to incorporate the microstructural size and to account for non uniform macroscopic deformation fields within the microstructural cell every microstructural constituent is modelled as a classical continuum and the rve problem is formulated in terms of standard equilibrium and boundary conditions from the solution of the microstructural boundary value problem the macroscopic stress tensor and the higher order stress tensor are derived based on an extension of the hill mandel condition this automatically delivers the microstructurally based constitutive response of the higher order macro continuum and deals with the microstructural size in a natural way several examples illustrate the approach particularly the microstructural size effects

finite element concepts for finite elastoplastic strains and isotropic stress response in shells theoretical and computational analysis this paper presents appropriate finite element concepts accounting for finite elastoplastic strains and isotropic stress response in arbitrary shells three parameterization strategies for the calculation of thin and thick smooth shells as well as shell intersections are discussed in detail in this context the importance of the deformation gradient for an efficient implementation is especially emphasized according to the parameterizations the computational plasticity algorithms are derived in general three dimensional 3 d form for thick shell structures and a restricted 2 d form satisfying the plane stress state underlying thin shells in order to describe arbitrary shell geometries the isoparametric concept is applied to all proposed finite element formulations these are based on quadrilateral 4 node mixed finite shell elements the variational approaches of the enhanced assumed strain method the assumed natural strain concept and the reduced integration technique are accurately considered by means of representative numerical examples the performance of the three different finite element concepts is demonstrated furthermore their individual properties are addressed with respect to efficiency and numerical accuracy thus crucial consequences for an optimal modelling of finite plastic strains in shells can be outlined

a theoretical and computational model for isotropic elastoplastic stress analysis in shells at large strains the article presents new aspects of large strain isotropic elastoplastic analysis of shells on the shell theory side we consider several possible continuum based parametrizations of the shell deformation and investigate their applicability with respect to the implementation of general three dimensional local constitutive models as an aspect of computational shell analysis we then discuss in detail the numerical implementation of a shell based on the parametrization of the top and bottom surface in terms of a new 8 node brick type mixed finite shell element on the basis of assumed strain and enhanced strain variational approaches on the side of plasticity theory we adopt a recently proposed intermediate configuration free formulation of large strain plasticity based on the notion of a plastic metric and consider its representation with respect to the parameter manifold of the shell this frame is formulated in terms of principal strains and principal stresses with respect to dual co and contra variant eigenvector triads associated with a mixed variant elastic strain tensor the new aspect of computational plasticity is the development of a stress update algorithm for the plasticity model mentioned above which includes a general return mapping in the eigenvalue space the algorithm can be applied to general coordinate charts and is particularly convenient when formulated with respect to the space of the curvilinear coordinates of the shell parametrization we demonstrate the performance of the finite element formulation and constitutive algorithms by means of several numerical examples

linear finite elements may be only first order pointwise accurate on anisotropic triangulations we give a counterexample of an anisotropic triangulation on which the exact solution has a second order error of linear interpolation while the computed solution obtained using linear finite elements is only first order pointwise accurate our example is given in the context of a singularly perturbed reaction diffusion equation whose exact solution exhibits a sharp boundary layer furthermore we give a theoretical justification of the observed numerical phenomena using a finite difference representation of the considered finite element methods both standard and lumped mass cases are addressed

energy norm a posteriori error estimates for singularly perturbed reaction diffusion problems on anisotropic meshes residual type a posteriori error estimates in the energy norm are given for singularly perturbed semilinear reaction diffusion equations posed in polygonal domains linear finite elements are considered on anisotropic triangulations the error constants are independent of the diameters and the aspect ratios of mesh elements and of the small perturbation parameter to deal with anisotropic triangulations a special quasi interpolation operator is employed that may be of independent interest

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

a singularly perturbed semilinear reaction diffusion problem in a polygonal domain the semilinear reaction di usion equation i 24u b x u 0 with dirichlet bound ary conditions is considered in a convex polygonal domain the singular perturbation parameter e is arbitrarily small and the reduced equation b x u0 x 0 may have multiple solutions an asymptotic expansion for u is constructed that involves boundary and corner layer functions by perturbing this asymptotic expansion we obtain certain sub and super solutions and thus show the existence of a solution u that is close to the constructed asymptotic expansion the polygonal boundary forces the study of the nonlinear autonomous elliptic equation dz f z 0 posed in an infinite sector and then well posedness of the corresponding linearized problem

three dimensional extension of non linear shell formulation based on the enhanced assumed strain concept conventional shell formulations such as 3 or 5 parameter theories or even 6 parameter theories including the thickness change as extra parameter require a condensation of the constitutive law in order to avoid a significant error due to the assumption of a linear displacement field across the thickness this means that the normal stress in thickness direction has to either vanish or be constant in general these extra constraints cannot be satisfied explicitly or they lead to elaborate strain expressions the main objective of the present study is to introduce directly a complete 3 d constitutive law without modification therefore a 7 parameter theory is utilized which includes a linear varying thickness stretch as extra variable allowing also large strain effects

uniqueness of solutions of parabolic semilinear nonlocal boundary problems abstract the purpose of the paper is to give two theorems about the uniqueness of solutions of parabolic semilinear nonlocal boundary problems the paper is a continuation of previous papers by byszewski and the generalization of some results from r rabczuk introduction to differential inequalities pwn warsaw 1976 polish j chabrowski on nonlocal problems for parabolic equations nagoya math j 93 1984 109 131 the theorems obtained in this paper can be applied in the theories of diffusion and heat conduction with better effects than the analogous theorems about parabolic initial boundary problems and than the analogous theorems about parabolic periodic boundary problems