a constitutive frame of elastoplasticity at large strains based on the notion of a plastic metric abstract the article presents a constitutive framework of large strain elastoplasticity in both the lagrangian and the eulerian geometric setting which takes into account anisotropic material response in summary the key ingredients of this framework are i the introduction of a plastic metric which is assumed to describe locally the history dependent inelastic material response in the sense of an internal variable formulation ii 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 iii an equivalent lagrangian and eulerian representation of all constitutive functions as isotropic tensor functions in terms of an extended set of arguments denoted as anisotropy variables iv the set up of normality rules for the evolution of the plastic metric and the anisotropy variables yielding a canonical symmetric form of the elastoplastic tangent moduli v a geometrically exact decomposition of the set of constitutive equations into possibly decoupled volumetric and isochoric contributions applications of the constitutive framework are demonstrated by means of several conceptual model problems which cover isotropic initial anisotropic and induced anisotropic elastoplastic response

discontinuous displacement approximation for capturing plastic localization it is proposed to capture localized plastic deformation via the inclusion of regularized displacement discontinuities at element boundaries interfaces of the finite element subdivision the regularization is based on a kinematic assumption for an interface that resembles that which is pertinent to the classical shear band concept as a by product of the regularization an intrinsic band width is introduced as a constitutive property rather than a geometric feature of the finite element mesh in this way the spurious mesh sensitivity which is obtained when the displacement approximation is continuous can be avoided another consequence is that the interfacial relation between the elements is derived directly from the conventional constitutive properties of the continuously deforming material an interesting feature is that the acoustic tensor will not only play a role for diagnosing discontinuous bifurcation but will also serve as the tangent stiffness tensor of the interface up to within a scalar factor an analytical investigation of the behaviour of the interface is carried out and it is shown that dilatation may indeed accompany slip within a shear band for a general plasticity model the significance of proper mesh alignment is demonstrated for a simple problem in plane strain and plane stress it is shown that a unique structural post peak response in accordance with non linear fracture mechanics can be achieved when the plastic softening modulus is properly related to the bandwidth the paper concludes with a numerical simulation of the gradual development of a shear band in a soil slope

analysis of microstructure development in shearbands by energy relaxation of incremental stress potentials large strain theory for standard dissipative solids we propose a fundamentally new approach to the treatment of shearband localizations in strain softening elastic plastic solids at finite strains based on energy minimization principles associated with microstructure developments the point of departure is a general internal variable formulation that determines the finite inelastic response as a standard dissipative medium consistent with this type of inelasticity we consider an incremental variational formulation of the local constitutive response where a quasi hyperelastic stress potential is obtained from a local constitutive minimization problem with respect to the internal variables the existence of this variational formulation allows the definition of the material stability of an inelastic solid based on weak convexity conditions of the incremental stress potential in analogy to treatments of finite elasticity furthermore localization phenomena are interpreted as microstructure developments on multiple scales associated with non convex incremental stress potentials in analogy to elastic phase decomposition problems these microstructures can be resolved by the relaxation of non convex energy functionals based on a convexification of the stress potential the relaxed problem provides a well posed formulation for a mesh objective analysis of localizations as close as possible to the non convex original problem based on an approximated rank one convexification of the incremental stress potential we develop a computational two scale procedure for a mesh objective treatment of localization problems at finite strains it constitutes a local minimization problem for a relaxed incremental stress potential with just one scalar variable representing the intensity of the microshearing of a rank one laminate aligned to the shear band this problem is sufficiently robust with regard to applications to large scale inhomogeneous deformation processes of elastic plastic solids the performance of the proposed energy relaxation method is demonstrated for a representative set of numerical simulations of straight and curved shear bands which report on the mesh independence of the results copyright 2003 john wiley sons ltd

numerical approximation of young measures in non convex variational problems summary in non convex optimisation problems in particular in non convex variational problems there usually does not exist any classical solution but only generalised solutions which involve young measures in this paper first a suitable relaxation and approximation theory is developed together with optimality conditions and then an adaptive scheme is proposed for the efficient numerical treatment the young measures solving the approximate problems are usually composed only from a few atoms this is the main argument our effective active set type algorithm is based on the support of those atoms is estimated from the weierstrass maximum principle which involves a hamiltonian whose good guess is obtained by a multilevel technique numerical experiments are performed in a one dimensional variational problem and support efficiency of the algorithm

localisation in a cosserat continuum under static and dynamic loading conditions abstract localisation studies have been carried out for an unconstrained elasto plastic strain softening cosserat continuum because of the presence of an internal length scale in this continuum model a perfect convergence is found upon mesh refinement a finite constant width of the localisation zone and a finite energy dissipation are computed under static as well as under transient loading conditions because of the existence of rotational degrees of freedom in a cosserat continuum additional wave types arise and wave propagation becomes dispersive this has been investigated analytically and numerically for an elastic cosserat continuum and an excellent agreement has been found between both solutions

strain driven homogenization of inelastic microstructures and composites based on an incremental variational formulation the paper investigates computational procedures for the treatment of a homogenized macro continuum with locally attached micro structures of inelastic constituents undergoing small strains the point of departure is a general internal variable formulation that determines the inelastic response of the constituents of a typical micro structure as a generalized standard medium in terms of an energy storage and a dissipation function consistent with this type of inelasticity we develop a new incremental variational formulation of the local constitutive response where a quasi hyperelastic micro stress potential is obtained from a local minimization problem with respect to the internal variables it is shown that this local minimization problem determines the internal state of the material for finite increments of time we specify the local variational formulation for a setting of smooth single surface inelasticity and discuss its numerical solution based on a time discretization of the internal variables the existence of the quasi hyperelastic stress potential allows the extension of homogenization approaches of elasticity to the incremental setting of inelasticity focusing on macro strain driven micro structures we develop a new incremental variational formulation of the global homogenization problem where a quasi hyperelastic macro stress potential is obtained from a global minimization problem with respect to the fine scale displacement fluctuation field it is shown that this global minimization problem determines the state of the micro structure for finite increments of time we consider three different settings of the global variational problem for prescribed linear displacements periodic fluctuations and constant stresses on the boundary of the micro structure and discuss their numerical solutions based on a spatial discretization of the fine scale displacement fluctuation field the performance of the proposed methods is demonstrated for the model problem of von mises type elasto visco plasticity of the constituents and applied to a comparative study of micro to macro transitions of inelastic composites copyright 2002 john wiley sons ltd

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

variational principles in dissipative electro magneto mechanics a framework for the macro modeling of functional materials summary this paper presents a general framework for the macroscopic continuum based formulation and numerical implementation of dissipative functional materials with electro magneto mechanical couplings based on incremental variational principles we focus on quasi static problems where mechanical inertia effects and time dependent electro magnetic couplings are a priori neglected and a time dependence enters the formulation only through a possible rate dependent dissipative material response the underlying variational structure of non reversible coupled processes is related to a canonical constitutive modeling approach often addressed to so called standard dissipative materials it is shown to have enormous consequences with respect to all aspects of the continuum based modeling in macroscopic electro magnetomechanics at first the local constitutive modeling of the coupled dissipative response i e stress electric and magnetic fields versus strain electric displacement and magnetic induction is shown to be variational based governed by incremental minimization and saddle point principles next the implications on the formulation of boundary value problems are addressed which appear in energy based formulations as minimization principles and in enthalpy based formulations in the form of saddle point principles furthermore the material stability of dissipative electro magneto mechanics on the macroscopic level is defined based on the convexity concavity of incremental potentials we provide a comprehensive outline of alternative variational structures and discuss details of their computational implementation such as formulation of constitutive update algorithms and finite element solvers from the viewpoint of constitutive modeling including the understanding of the stability in coupled electro magneto mechanics an energy based formulation is shown to be the canonical setting from the viewpoint of the computational convenience an enthalpy based formulation is the most convenient setting a numerical investigation of a multiferroic composite demonstrates perspectives of the proposed framework with regard to the future design of new functional materials copyright 2011 john wiley sons ltd received 1 october 2010 revised 15 december 2010 accepted 15 december 2010

accuracy of two svd algorithms for 2 2 triangular matrices a new algorithm for the accurate computation of the singular value decomposition of 2x2 triangular matrices is proposed the algorithm is based on voevodin formulas sharp accuracy bounds are derived by using a subtle error analysis which tracks the signs of the errors of intermediate quantities and does not neglect the non linear parts of the errors the analysis is fine tuned for the case of almost diagonal matrices the same analysis is also used to analyze the errors for the xlasv2 computational routine of lapack the error estimates of the new algorithm compare favorably to those of the lapack routine

an analysis of strong discontinuities in multiplicative finite strain plasticity and their relation with the numerical simulation of strain localization in solids abstract this paper presents an analysis of strong discontinuities in inelastic solids at finite strains solutions exhibiting this type of discontinuities characterized by a discontinuous displacement field are shown to make physical and mathematical sense in a classical multiplicative plasticity continuum model if the softening modulus is reinterpreted as a singular distribution physically the strainsoftening is localized along the discontinuity conditions for the appearance of strong discontinuities in the geometrically nonlinear range are characterized as it is the response of the material during localization in addition these analytical results are exploited in the design of a new class of finite element methods the proposed methods fall within the class of enhanced strain methods and lead to solutions independent of the mesh size and insensitive to mesh alignment without requiring any regularization of the solutions by numerical parameters like a characteristic length

geometrically non linear enhanced strain mixed methods and the method of incompatible modes a class of assumed strain mixed finite element methods for fully non linear problems in solid mechanics is presented which when restricted to geometrically linear problems encompasses the classical method of incompatible modes as a particular case the method relies crucially on a local multiplicative decomposition of the deformation gradient into a conforming and an enhanced part formulated in the context of a three field variational formulation the resulting class of mixed methods provides a possible extension to the non linear regime of well known incompatible mode formulations in addition this class of methods includes non linear generalizations of recently proposed enhanced strain interpolations for axisymmetric problems which cannot be interpreted as incompatible modes elements the good performance of the proposed methodology is illustrated in a number of simulations including 2 d 3 d and axisymmetric finite deformation problems in elasticity and elastoplasticity remarkably these methods appear to be specially well suited for problems involving localization of the deformation as illustrated in several numerical examples

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

reliable and efficient approximation of polyconvex envelopes an iterative algorithm that approximates the polyconvex envelope fpc of a given function f mathbb r n times m to mathbb r i e the largest function below f which is convex in all minors is established also presented are a rigorous error analysis with a focus on reliability and optimal orders of convergence an efficient strategy that reduces the large number of unknowns as well as numerical experiments

semicontinuity problems in the calculus of variations soit f r 3 r une fonction deux fois continument differentiable et pour u w 1 1 a b f u a b a b f x u x u x dx si f r n r m r nm r satisfait 1 f est mesurable en x continue en cs et 2 0 f x s a x c s p p ou p 1 a est une fonction localement sommable non negative et c une constante non negative alors la fonctionnelle f u a b est sequentiellement faiblement semi continue inferieurement sur w 1 p r m si et seulement si f x s est quasi convexe pour chaque s et presque chaque x

a novel algorithmic framework for the numerical implementation of locally embedded strong discontinuities in this paper a novel algorithmic framework for the numerical modeling of locally embedded strong discontinuities suitable for the analysis of material failure such as cracking in brittle structures or shear bands in soils is proposed based on the enhanced assumed strain eas concept the final failure kinematics of solids approximated by discontinuous displacement fields are incorporated into the finite element formulation in contrast to previous works the discontinuous part of the deformation mapping is condensed out at the material level without employing the standard static condensation technique as a consequence the resulting constitutive equations are formally identical to those of standard plasticity models hence the return mapping algorithm is applied to the numerical implementation only slight modifications of this by now classical algorithm are necessary the proposed framework is applicable to any constitutive equation characterizing the inelastic part of deformation referring to the yield failure function and the evolution equations no special assumption has to be made despite the differences between the presented numerical framework and the original implementation of the strong discontinuity approach sda it is shown that both finite element models are completely equivalent the applicability of the novel algorithmic formulation and its numerical performance are investigated by means of two two dimensional as well as by means of a fully three dimensional numerical analysis of shear band formation

computational micro to macro transitions for discretized micro structures of heterogeneous materials at finite strains based on the minimization of averaged incremental energy the paper investigates algorithms for the computation of homogenized stresses and overall tangent moduli of micro structures undergoing large strain deformations typically these micro structures define representative volumes of nonlinear heterogeneous materials such as inelastic composites polycrystalline aggregates or particle assemblies we consider a priori given discretized micro structures without focussing on details of specific discretization techniques for space and time the key contribution of the paper is the construction of a family of algorithms and matrix representations of overall properties of discretized micro structures which are motivated by a minimization of averaged incremental energy it is shown that the overall stresses and tangent moduli of a typical micro structure may exclusively be defined in terms of discrete forces and stiffness properties on the boundary we focus on deformation driven micro structures where the overall macroscopic deformation is controlled in this context three classical types of boundary conditions are investigated i linear deformation ii uniform tractions and iii periodic deformation and antiperiodic tractions incorporated by lagrangian multiplier methods these conditions generate three classes of constrained minimization problems with associated solution algorithms for the computation of equilibrium states and overall properties of micro structures the proposed algorithms and matrix representations of the overall properties are formally independent of the interior spatial structure and the local constitutive response of the micro structure and are therefore applicable to a broad class of model problems we demonstrate their performance for some representative model problems including finite elastic plastic deformations of composites texture developments in polycrystalline materials and equilibrium states of particle assemblies

a variational multiscale approach to strain localization formulation for multidimensional problems abstract the multiscale approach to strain localization problems developed earlier is generalized to higher dimensions the key idea is the identification of the fine scale field with a component of the displacement that has a large gradient the difference between the total displacement and the fine scale is the coarse scale field the formulation is variationally based with the weak form as the point of departure with suitable assumptions on the space of fine scale fields the weak form is separable into coarse and fine scale parts the weak form of the fine scale problem is used to eliminate the fine scales from the formulation this is achieved by way of a projection onto the fine scale space and allows a reformulation of the full problem in terms of coarse scale components alone in this framework fine scale interpolations are identified that ensure sharp resolution of localized displacement irrespective of the underlying finite element mesh it is demonstrated that by correctly accounting for the microstructural fine scale behavior the formulation results in solutions devoid of pathological mesh sensitivity for problems wherein the detailed structure of the localized displacement is of interest recovery of the fine scale components can be performed to reconstruct the entire displacement field several numerical examples are presented that demonstrate the robustness of the formulation

numerical solution of the scalar double well problem allowing microstructure the direct numerical solution of a non convex variational problem p typically faces the difficulty of the finite element approximation of rapid oscillations although the oscillatory discrete minimisers are properly related to corresponding young measures and describe real physical phenomena they are costly and difficult to compute in this work we treat the scalar double well problem by numerical solution of the relaxed problem rp leading to a degenerate convex minimisation problem the problem rp has a minimiser u and a related stress field dw u which is known to coincide with the stress field obtained by solving p in a generalised sense involving young measures if u h is a finite element solution h dw u h is the related discrete stress field we prove a priori and a posteriori estimates for h in l 4 3 and weaker weighted estimates for u u h the a posteriori estimate indicates an adaptive scheme for automatic mesh refinements as illustrated in numerical experiments

on the relation between the principle of maximum dissipation and inelastic evolution given by dissipation potentials we study the evolution of systems described by internal variables after the introduction of thermodynamic forces and fluxes both the dissipation and dissipation potential are defined then the principle of maximum dissipation pmd and a minimum principle for the dissipation potential are developed in a variational formulation both principles are related to each other several cases are shown where both principles lead to the same evolution equations for the internal variables however also counterexamples are reported where such an equivalence is not valid in this case an extended pmd can be formulated

optimal design and relaxation of variational problems iii abstract this paper has attempted a synthesis of three subjects that have developed separately optimal design homogenization and relaxation in the calculus of variations the underlying theme is that of oscillations they occur with certain definite patterns in minimizing sequences for the variational problem the same patterns recur in nearly optimal designs and in the coefficients of the equation being homogenized each problem is resolved by determining the structure of those oscillations and above all their macroscopic averages it is those averages that solve the relaxed problem they also determine the optimal design and the solution of the homogenized equation we emphasize also the role of optimization the homogenized equations and composite materials of interest here come not from arbitrary mixtures but from those that are extremal since they correspond to minimizing sequences for the variational problem the effective properties of these composites attain the optimal bounds for given materials in fixed proportions

deriving new evolution equations for microstructures via relaxation of variational incremental problems we study material models for rate independent inelasticity in situations where no internal length scale is given and formation of microstructure for the deformation gradients and the internal variables may occur we develop a rational procedure for deriving consistent macroscopic models which allow for the computation of nontrivial effective quantities without resolving the fine scales the method involves the relaxation of variational incremental problems which are derived from an elastic and a plastic potential we use young measures to describe the microscopic distribution of the internal variables as well as the quasiconvexification of the elastic stored energy density with respect to the deformation gradient the resulting model provides a new rate independent model in terms of the deformation and the young measure the approach is based on a derivative free energetic formulation using one functional for elastic energy storage and one for the dissipation distance the latter is derived from a dissipation potential defining the dissipation distance between internal states and hence the wasserstein distance between young measures this approach is strongly linked to an associated time incremental problem which is a minimization problem of the type used for several years now in the engineering literature the update algorithm for the incremental problem is discussed in detail and two simple examples are given

mathbf h principles for hypersurfaces with prescribed principle curvatures and directions we prove that any compact orientable hypersurface with boundary immersed resp embedded in euclidean space is regularly homotopic resp isotopic to a hypersurface with principal directions which may have any prescribed homotopy type and principal curvatures each of which may be prescribed to within an arbitrary small error of any constant further we construct regular homotopies resp isotopies which control the principal curvatures and directions of hypersurfaces in a variety of ways these results which we prove by holonomic approximation establish some h principles in the sense of gromov and generalize theorems of gluck and pan on embedding and knotting of positively curved surfaces in 3 space

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

implementation of incremental variational formulations based on the numerical calculation of derivatives using hyper dual numbers abstract in this paper novel implementation schemes for the automatic calculation of internal variables stresses and consistent tangent moduli for incremental variational formulations ivfs describing inelastic material behavior are proposed ivfs recast inelasticity theory as an equivalent optimization problem where the incremental stress potential within a discrete time interval is minimized in order to obtain the values of internal variables in the so called multilevel newton raphson method for the inelasticity theory this minimization problem is typically solved by using second derivatives with respect to the internal variables in addition to that to calculate the stresses and moduli further second derivatives with respect to deformation tensors are required compared with classical formulations such as the return mapping method the ivfs are relatively new and their implementation is much less documented furthermore higher order derivatives are required in the algorithms demanding increased implementation efforts therefore even though ivfs are mathematically and physically elegant their application is not standard here novel approaches for the implementation of ivfs using hdns of second and higher order are presented to arrive at a fully automatic and robust scheme with computer accuracy the proposed formulations are quite general and can be applied to a broad range of different constitutive models which means that once the proposed schemes are implemented as a framework any other dissipative material model can be implemented in a straightforward way by solely modifying the constitutive functions these include the helmholtz free energy function the dissipation potential function and additional side constraints such as e g the yield function in the case of plasticity its uncomplicated implementation for associative finite strain elasto plasticity and performance is illustrated by some representative numerical examples

non convex potentials and microstructures in finite strain plasticity a mathematical model for a finite strain elastoplastic evolution problem is proposed in which one time step of an implicit time discretization leads to generally non convex minimization problems the elimination of all internal variables enables a mathematical and numerical analysis of a reduced problem within the general framework of calculus of variations and nonlinear partial differential equations the results for a single slip system and von mises plasticity illustrate that finite strain elastoplasticity generates reduced problems with non quasiconvex energy densities and so allows for non attainment of energy minimizers and microstructures

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

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

on a class of constitutive equations in viscoplasticity formulation and computational issues the viscoplastic constitutive model is formulated based on the existence of the dissipation potential which embodies the notion of the gauge minkowski function of the convex set a perturbation method is used for a solution of stiff differential equations characterizing the associated problem of evolution it relies on a discrete formulation of viscoplasticity which results from the regularized version of the principle of maximum plastic dissipation the operator split methodology and the newton raphson method are used to obtain the numerical solution of the discretized equations of evolution the consistent tangent modulus is expressed in a closed form as a result of the exact linearization of the discretized evolution equations for several variants of the flow potential function including some representative stiff functional forms numerical tests of the integration algorithm based on iso error maps are provided finally a numerical example is presented to illustrate the robustness and the effectiveness of the proposed approach

the variational formulation of viscoplastic constitutive updates we present a class of constitutive updates for general viscoplastic solids including such aspects of material behavior as finite elastic and plastic deformations non newtonian viscosity rate sensitivity and arbitrary flow and hardening rules the distinguishing characteristic of the proposed constitutive updates is that by construction the corresponding incremental stress strain relations derive from a pseudo elastic strain energy density this in turn confers the incremental boundary value problem a variational structure in particular the incremental deformation mapping follows from a minimum principle this minimum principle may conveniently be taken as a basis for error estimation and mesh adaption the accuracy and robustness of the variational constitutive updates is demonstrated with the aid of convergence tests involving the finitely deforming mises solid and ductile single crystals the ability of the updates to resolve the complex patterns of slip activity which arise in the latter application is particularly noteworthy