subdivision based mixed methods for isogeometric analysis of linear and nonlinear nearly incompressible materials this paper addresses the use of isogeometric analysis to solve solid mechanics problems involving nearly incompressible materials the present work is focused on extension of two field mixed variational formulations in both small and large strains to isogeometric analysis inf sup stable displacement pressure combinations for mixed formulations are developed based on the subdivision property of nurbs stability and convergence properties of the proposed displacement pressure combinations are illustrated by computing numerical inf sup constants and error norms the performance of the proposed formulations is assessed by studying several benchmark examples involving nearly incompressible and incompressible elastic and elasto plastic materials in both small and large strain regime

exponential decay of correlations for strongly coupled toom probabilistic cellular automata we investigate the low noise regime of a large class of probabilistic cellular automata including the north east center model of toom they are defined as stochastic perturbations of cellular automata belonging to the category of monotonic binary tessellations and possessing a property of erosion we prove for a set of initial conditions exponential convergence of the induced processes toward an extremal invariant measure with a highly predominant spin value we also show that this invariant measure presents exponential decay of correlations in space and in time and is therefore strongly mixing

smooth finite strain plasticity with non local pressure support the aim of this work is to introduce an alternative framework to solve problems of finite strain elasto plasticity including anisotropy and kinematic hardening coupled with any isotropic hyperelastic law after deriving the constitutive equations and inequalities without any of the customary simplifications we arrive at a new general elasto plastic system we integrate the elasto plastic algebraico differential system and replace the loading unloading condition by a chen mangasarian smooth function to obtain a non linear system solved by a trust region method despite being non standard this approach is advantageous since quadratic convergence is always obtained by the non linear solver and very large steps can be used with negligible effect in the results discretized equilibrium is in contrast with traditional approaches smooth and well behaved in addition since no return mapping algorithm is used there is no need to use a predictor the work follows our previous studies of element technology and highly non linear visco elasticity from a general framework with exact linearization systematic particularization is made to prototype constitutive models shown as examples our element with non local pressure support is used examples illustrating the generality of the method are presented with excellent results

exponential convergence of toom s probabilistic cellular automata we obtain cluster expansions for small random perturbations of deterministic toom s automata in the one dimensional case exponential convergence follows analyticity of invariant measures is examined as well as the simplest multidimensional case

finite deformation higher order shell models and rigid body motions this paper focuses on the developments of higher order shell models by employing the new concept of interpolation surfaces i surfaces inside the shell body we introduce n n p 3 i surfaces and choose the values of displacements with correspondence to these surfaces as fundamental shell unknowns such choice allows first to develop various higher order shell models in a very compact form and second to derive non linear strain displacement relationships which are completely free for arbitrarily large rigid body motions the general 3n parameter shell model is proposed in the framework of the lagrangian description the special 9 12 and 15 parameters cases corresponding to n 3 4 and 5 have been dealt in detail the proposed shell models account for thickness stretching and the complete 3d constitutive equations are utilized the displacement vectors of equally located i surfaces for each model are represented in a convected curvilinear coordinate system allowing one to develop directly finite deformation geometrically exact shell finite elements which could be discussed in future works 2008 elsevier ltd all rights reserved

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

phase transition and correlation decay in coupled map lattices for a coupled map lattice with a specific strong coupling emulating stavskaya s probabilistic cellular automata we prove the existence of a phase transition using a peierls argument and exponential convergence to the invariant measures for a wide class of initial states using a technique of decoupling originally developed for weak coupling this implies the exponential decay in space and in time of the correlation functions of the invariant measures

phase diagrams of majority voter probabilistic cellular automata we present a detailed numerical analysis of the phase diagrams for some majority voter probabilistic cellular automata and connect the results with theory enabling us to prove many of the observed features

small displacements about equilibrium of a beam subjected to large static loads small vibrations about the equilibrium configuration of a beam structure are treated the beam is subjected to large static loads causing large displacements the influence of these static loads on the small vibrations is investigated in the stationary harmonic case the symmetry of the dynamic stiffness matrix is discussed a semifollower static moment is introduced to maintain the symmetry of the dynamic stiffness matrix and to create conservative subsystems the beam is deformed only in bending and torsion i e no extension of the geometric centerline is permitted and no transverse shear of the beam lamina is taken into account the euler bernoulli beam theory is applied to bending and the saint venant theory to torsion an euler axis transformation is used for handling the large rotations in the nonlinear case the material of the beam is assumed to be linearly elastic a continuous model is made of the beam i e the governing differential equations are derived and solved without recourse to a spatial discretization

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

uniqueness of the srb measure for piecewise expanding weakly coupled map lattices in any dimension we prove the existence of a unique srb measure for a wide range of multidimensional weakly coupled map lattices these include piecewise expanding maps with diffusive coupling

four node semi eas element in six field nonlinear theory of shells we propose a new four node c finite element for shell structures undergoing unlimited translations and rotations the considerations concern the general six field theory of shells with asymmetric strain measures in geometrically nonlinear static problems the shell kinematics is of the two dimensional cosserat continuum type and is described by two independent fields the vector field for translations and the proper orthogonal tensor field for rotations all three rotational parameters are treated here as independent hence as a consequence of the shell theory the proposed element has naturally six engineering degrees of freedom at each node with the so called drilling rotation this property makes the element suitable for analysis of shell structures containing folds branches or intersections to avoid locking phenomena we use the enhanced assumed strain eas concept we derive and linearize the modified hu washizu principle for six field theory of shells what makes the present approach original is the combination of eas method with asymmetric membrane strain measures based on literature we propose new enhancing field and specify the transformation matrix that accounts for the lack of symmetry to gain knowledge about the suitability of this field for asymmetric strain measures and to assess the performance of the element we solve typical benchmark examples with smooth geometry and examples involving orthogonal intersections of shell branches

statistical mechanics of probabilistic cellular automata we investigate the behavior of discrete time probabilistic cellular automata pca which are markov processes on spin configurations on ad dimensional lattice from a rigorous statistical mechanics point of view in particular we exploit whenever possible the correspondence between stationary measures on the space time histories of pcas on d and translation invariant gibbs states for a related hamiltonian on d 1 this leads to a simple large deviation formula for the space time histories of the pca and a proof that in a high temperature regime the stationary states of the pca are gibbsian we also obtain results about entropy fluctuations and correlation inequalities and demonstrate uniqueness of the invariant state and exponential decay of correlations in a high noise regime we discuss phase transitions in the low noise or low temperature regime and review toom s proof of nonergodicity of a certain class of pcas

genuinely resultant shell finite elements accounting for geometric and material non linearity the underlying theory is statically and geometrically exact and it naturally includes small strain and finite strain problems of thin as well as thick shells this paper examines the development of computational procedures employing the newton kantorovich linearization process and the galerkin type discretization method the treatment of finite rotations through an arbitrary parametrization of the rotation group the interpolation procedure of so 3 valued functions underlying the construction of finite element basis

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

a finite strain quadrilateral shell element based on discrete kirchhoff love constraints this paper improves the 16 degrees of freedom quadrilateral shell element based on pointwise kirchhoff love constraints and introduces a consistent large strain formulation for this element the model is based on classical shell kinematics combined with continuum constitutive laws the resulting element is valid for large rotations and displacements the degrees of freedom are the displacements at the corner nodes and one rotation at each mid side node the formulation is free of enhancements it is almost fully integrated and is found to be immune to locking or unstable modes the patch test is satisfied in addition the formulation is simple and amenable to efficient incorporation in large scale codes as no internal degrees of freedom are employed and the overall calculations are very efficient results are presented for linear and non linear problems copyright 2005 john wiley sons ltd

non linear dynamics of three dimensional rods exact energy and momentum conserving algorithms the long term dynamic response of non linear geometrically exact rods under going finite extension shear and bending accompanied by large overall motions is addressed in detail the central objective is the design of unconditionally stable time stepping algorithms which exactly preserve fundamental constants of the motion such as the total linear momentum the total angular momentum and for the hamiltonian case the total energy this objective is accomplished in two steps first a class of algorithms is introduced which conserves linear and angular momentum this result holds independently of the definition of the algorithmic stress resultants second an algorithmic counterpart of the elastic constitutive equations is developed such that the law of conservation of total energy is exactly preserved conventional schemes exhibiting no numerical dissipation symplectic algorithms in particular are shown to lead to unstable solutions when the high frequencies are not resolved compared to conventional schemes there is little if any additional computational cost involved in the proposed class of energy momentum methods the excellent performance of the new algorithm in comparison to other standard schemes is demonstrated in several numerical simulations

unilateral non linear dynamic contact of thin walled structures using a primal dual active set strategy the efficient modelling of three dimensional contact problems is still a challenge in non linear implicit structural analysis we use a primal dual active set strategy siam j optim 2003 13 865 888 based on dual lagrange multipliers siam j numer anal 2000 38 989 1012 to handle the non linearity of the contact conditions this allows us to enforce the contact constraints in a weak integral sense without any additional parameter due to the biorthogonality condition of the basis functions the lagrange multipliers can be locally eliminated we perform a static condensation to achieve a reduced system for the displacements the lagrange multipliers representing the contact pressure can be easily recovered from the displacements in a variationally consistent way for the application to thin walled structures we adapt a three dimensional non linear shell formulation including the thickness stretch of the shell to contact problems a reparametrization of the geometric description of the shell body gives us a surface oriented shell element which allows the application of contact conditions directly to nodes lying on the contact surface shell typical locking phenomena are treated with the enhanced assumed strain method and the assumed natural strain method the discretization in time is done with the implicit generalized method j appl mech 1993 60 371 375 and the generalized energy momentum method comp methods appl mech eng 1999 178 343 366 to compare the development of energies within a frictionless contact description in order to conserve the total energy within the discretized frictionless contact framework we follow an approach from laursen and love int j numer methods eng 2002 53 245 274 who introduced a discrete contact velocity to update the velocity field in a post processing step various examples show the good performance of the primal dual active set strategy applied to the implicit dynamic analysis of thin walled structures

finite strain fracture of plates and shells with configurational forces and edge rotations we propose a simple and efficient algorithm for fem based computational fracture of plates and shells cf 1 with both brittle and ductile materials based on edge rotation and load control rotation axes are the crack front nodes and each crack front edge in surface discretizations affects the position of only one or two nodes modified positions of the entities maximize the mesh quality complying with the predicted crack path which depends on the specific propagation theory in use compared with xfem or with classical tip remeshing the proposed solution has algorithmic and generality advantages the propagation algorithm is simpler than the aforementioned alternatives and the approach is independent of the underlying element used for discretization for history dependent materials there are still some transfer of relevant quantities between elements however diffusion of results is more limited than with tip or full remeshing to illustrate the advantages of our approach three prototype models are used tip energy dissipation lefm cohesive zone approaches and ductile fracture both the sutton crack path criterion and the path estimated by the eshelby tensor are employed traditional fracture benchmarks including one with plastic hinges and newly proposed verification tests are solved these were found to be very good in terms of crack path and load deflection accuracy

computational homogenisation of composite plates consideration of the thickness change with a modified projection strategy in the present paper a method for the modelling of the mechanical behaviour of composite plates especially hybrid laminates and sandwich plates is proposed the chosen method is the computational homogenisation or the so called fe 2 for plates the principle is to split the considered problem into two scales on the one hand the two dimensional fe computation of a plate is performed on the macroscale on the other hand a three dimensional fe problem is computed on the mesoscale which enables the discretisation of the layers organisation from each integration point of the macroscale the deformations are projected to the representative volume element rve on the mesoscale where a dirichlet boundary value problem is solved finally the homogenisation of the stresses is conducted to define the stress resultants of the macroscale the proposed method presents the advantage to take into account any material behaviour without any transformation of the constitutive laws even non linear ones furthermore the new projection strategy presented in this work considers the thickness change of the plate which enables a resolution of the poisson locking the method is illustrated by the simulation of the mechanical behaviour of both sandwich plate and of hybrid laminate

nonlinear finite element analysis of elastic systems under nonconservative loading natural formulation part ii dynamic problems abstract the paper presents a nonlinear finite element analysis of elastic structures subject to nonconservative forces attention is focused on the stability behaviour of such systems this leads mathematically to non self adjoint boundary value problems which are of great theoretical and practical interest in particular in connection with alternative modes of instability like divergence of flutter only quasistatic effects are however considered in the present part the methodology of our theory is general but the specific thrust of the present research is directed towards the analysis of structures acted upon by displacement dependent nonconservative follower forces in a finite element formulation the analysis of geometrically nonlinear elastic systems subject to such forces gives in general rise to a contributory nonsymmetric stiffness matrix known as the load correction matrix as a result the total tangent stiffness matrix becomes unsymmetric an indication of the non self adjoint character of the problem our theory is based on the natural mode technique 1 2 3 and permits i a a simple but elegant derivation of the load correction matrix the application of the general theory as evolved in this paper is demonstrated on the beam element in space a number of numerical examples are considered including divergence and flutter types of instability for which exact analytic solutions are known the problems demonstrate the efficiency of the present finite element formulation the paper furnishes also a novel and concise formulation of finite rotations in space which may be considered as a conceptual generalization of the theory presented in 2 3

multiple quadrature underintegrated finite elements new multiple quadrature point underintegrated finite elements with hourglass control are developed the elements are selectively underintegrated to avoid volumetric and shear locking and save computational time an approach for hourglass control is proposed such that the stabilization operators are obtained simply by taking the partial derivatives of the generalized strain rate vector with respect to the natural co ordinates so that the elements require no stabilization parameter to improve accuracy over the traditional one point quadrature elements several quadrature points are used to integrate the internal forces especially for tracing the plastic fronts in the mesh during loading and unloading in elastic plastic analysis two and four point quadrature elements are proposed for use in the two and three dimensional elements respectively other multiple quadrature points can also be employed several numerical examples such as thin beam plate and shell problems are presented to demonstrate the applicability of the proposed elements

a finite strain solid shell using local lowdin frames and least squares strains abstract a finite strain solid shell element is proposed it is based on least squares in plane assumed strains assumed natural transverse shear and normal strains the singular value decomposition svd is used to define local integration point orthogonal frames of reference solely from the jacobian matrix the complete finite strain formulation is derived and tested assumed strains obtained from least squares fitting are an alternative to the enhanced assumed strain eas formulations and in contrast with these the result is an element satisfying the patch test there are no additional degrees of freedom as it is the case with the enhanced assumed strain case even by means of static condensation least squares fitting produces invariant finite strain elements which are shear locking free and amenable to be incorporated in large scale codes with that goal we use automatically generated code produced by acegen and mathematica all benchmarks show excellent results similar to the best available shell and hybrid solid elements with significantly lower computational cost

relations between incompatible displacement model and hybrid stress model an examination of the variational formulations confirms the similarity between the incompatible displacement model and the assumed stress hybrid model that was pointed out by irons in 1972 but the basic differences between the two are also identified for 8 node solid elements the assumed stress terms obtained through a rational procedure also agree with those deduced by irons purely from his physical insight

a multiple quadrature eight node hexahedral finite element for large deformation elastoplastic analysis abstract a multiple quadrature underintegrated hexahedral finite element which is free of volumetric and shear locking and has no spurious singular modes is described and implemented for nonlinear analysis finite element formulations are derived in the corotational coordinate system the use of consistent tangent operators for large deformation elastoplasticity with nonlinear isotropic kinematic hardening rules preserves the quadratic rate of convergence of the newton s iteration method in static analysis test problems studied demonstrate the efficiency and effectiveness of this element in solving a wide variety of problems including sheet metal forming processes

effects of element distortions on the performance of isoparametric elements we discuss the effects of element distortions on the performance of displacement based isoparametric quadrilateral finite elements suitable example problems for both the lagrangian and the serendipity types of elements are used to show numerically the effects of element distortions the effects of angular and curved edge distortions are evaluated analytically in terms of the abilities of the elements to represent exactly various polynomial displacement fields we then solve a plane stress problem adaptively to demonstrate the effect of element distortions on the accuracy of the stress distribution the conclusion is that the lagrangian elements are not affected by angular distortions and are therefore the more reliable elements for general use

a model adaptive hanging node concept based on a new non linear solid shell formulation in this contribution we propose an innovative approach to automatically refine shell finite element meshes in thickness direction the procedure employs a new eight node solid shell element formulation based on the concept of reduced integration with hourglass stabilization apart from the important advantages of computational efficiency and robustness this element formulation offers the possibility to work with a variable mesh density in thickness direction in the present paper we suggest an adaptive approach using the hanging node concept to avoid additional mesh distortion among different stress and strain based error criteria the warping of the cross section and the residual error in the solution of the strong form of the balance of linear momentum have been proven to be suitable error measures the former represents the deviation of the realistically non linear dependence of the characteristic strain coefficient from its corresponding linear distribution applications show that the mesh refines as expected in loading domains in the neighbourhood of bearings or regions with strongly accumulating inelastic deformation on the other hand the discretization remains rather coarse in uncritical parts of the structure which decreases the computational effort noticeably

a robust non linear mixed hybrid quadrilateral shell element in the paper a non linear quadrilateral shell element for the analysis of thin structures is presented the variational formulation is based on a hu washizu functional with independent displacement stress and strain fields the interpolation matrices for the mid surface displacements and rotations as well as for the stress resultants and strains are specified restrictions on the interpolation functions concerning fulfillment of the patch test and stability are derived the developed mixed hybrid shell element possesses the correct rank and fulfills the in plane and bending patch test using newton s method the finite element approximation of the stationary condition is iteratively solved our formulation can accommodate arbitrary non linear material models for finite deformations in the examples we present results for isotropic plasticity at finite rotations and small strains as well as bifurcation problems and post buckling response the essential feature of the new element is the robustness in the equilibrium iterations it allows very large load steps in comparison to other element formulations copyright 2005 john wiley sons ltd

a quadrilateral mixed finite element with two enhanced strain modes summary an improved plane strain stress element is derived using a hu washizu variational formulation with bilinear displacement interpolation seven strain and stress terms and two enhanced strain modes the number of unknowns of the four node element is increased from eight to ten degrees of freedom for linear and non linear applications the two unknowns associated with the enhanced strain terms can be eliminated by static condensation so that eight displacement degrees of freedom remain for the proposed element which is denoted by qe2 the excellent performance of the proposed element is demonstrated using several linear and non linear examples

stationary states of toom cellular automata in simulations stationary states of toom probabilistic cellular automata are tested in computer experiments the aim of the tests is to identify the features characterizing the equilibrium states understood as in statistical mechanics namely we investigate the following scaling laws that involve critical parameters beta gamma and nu locality of the interactions and density of the relative entropy between stationary states the arguments showing that stationary toom states are not the equilibrium ones are provided

optimal solid shells for non linear analyses of multilayer composites ii dynamics abstract we are presenting a simple low order solid shell element formulation having only displacement degrees of freedom dofs i e without rotational dofs that has an optimal number of parameters to pass the patch tests and thus allows for efficient and accurate analyses of large deformable multilayer shell structures using elements at extremely high aspect ratio with the dynamics referred to a fixed inertial frame the elements can be used to analyze multilayer shell structures undergoing large overall motion the formulation of this element is based on the mixed hu washizu variational principle leading to a novel enhancing strain tensor enhanced assumed strain eas method that renders the computation particularly efficient with improved in plane and out of plane bending behavior poisson thickness locking especially in refined analyses of composite structures involving a large number of high aspect ratio layers the energy momentum conserving algorithm in the context of current solid shell element is presented we discuss the eas formulation based on the displacement gradient and its complexity compared to formulation on the green lagrange strain shear locking and curvature thickness locking are treated using the assumed natural strain ans method the element has an optimal combination of the ans method and the minimal number of eas parameters required to pass the plate bending patch test numerical examples involving dynamic analyses with conservation of energy and momentum of multilayer shell structures having a large range of element aspect ratios are presented several implicit direct integration methods with without numerical dissipation are used and compared in terms of the accuracy stability and cost in multilayer shell structure finally we note that the topic in this paper is a fitting dedication to professor ekkehard ramm who has made important pioneering contributions in this research direction

b and f projection methods for nearly incompressible linear and non linear elasticity and plasticity using higher order nurbs elements abstract this paper presents projection methods to treat the incompressibility constraint in small and large deformation elasticity and plasticity within the framework of isogeometric analysis after reviewing some fundamentals of isogeometric analysis we investigate the use of higher order non uniform rational b splines nurbs within the b projection method the higher continuity property of such functions is explored in nearly incompressible applications and shown to produce accurate and robust results a new non linear f projection method based on a modified minimum potential energy principle and inspired by the b method is proposed for the large deformation case it leads to a symmetric formulation for which the consistent linearized operator for fully non linear elasticity is derived and used in a newton raphson iterative procedure the performance of the methods is assessed on several numerical examples and results obtained are shown to compare favorably with other published techniques

rational approach for assumed stress finite elements a new method for the formulation of hybrid elements by the hellinger reissner principle is established by expanding the essential terms of the assumed stresses as complete polynomials in the natural coordinates of the element the equilibrium conditions are imposed in a variational sense through the internal displacements which are also expanded in the natural co ordinates the resulting element possesses all the ideal qualities i e it is invariant it is less sensitive to geometric distortion it contains a minimum number of stress parameters and it provides accurate stress calculations for the formulation of a 4 node plane stress element a small perturbation method is used to determine the equilibrium constraint equations the element has been proved to be always rank sufficient

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 framework for finite strain elastoplasticity based on maximum plastic dissipation and the multiplicative decomposition part ii computational aspects computational aspects of the formulation of elastoplasticity proposed in part i of this work employing a multiplicative decomposition and based on the principle of maximum plastic dissipation are examined in detail in particular the implementation of the proposed hyperelastic j2 flow theory is shown to reduce to a trivial modification of the classical radial return algorithm which is amenable to exact linearization this results in a closed form expression for the consistent elastoplastic tangent moduli the algorithmic treatment of the general case is considered subsequently it is emphasized that objective rates and incrementally objective algorithms play no role in the present approach a number of numerical experiments are presented that illustrate the excellent performance of the proposed formulation

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