a reader s guide to gacs s positive rates paper peter gacs s monograph which follows this article provides a counterexample to the important positive rates conjecture this conjecture which arose in the late 1960 s was based on very plausible arguments some of which come from statistical mechanics during the long gestation period of the gacs example there has been a great deal of skepticism about the validity of his work the construction and verification of gacs s counterexample are unavoidably complex and as a consequence his paper is quite lengthy but because of the novelty of the techniques and the significance of the result his work deserves to become widely known this reader s guide is intended both as a cheap substitute for reading the whole thing as well as a warm up for those who want to plumb its depths

stabilized finite element formulation for elastic plastic finite deformations abstract this paper presents a stabilized finite element formulation for nearly incompressible finite deformations in hyperelastic plastic solids such as metals an updated lagrangian finite element formulation is developed where mesh dependent terms are added to enhance the stability of the mixed finite element formulation this formulation circumvents the restriction on the displacement and pressure fields due to the babuska brezzi condition and provides freedom in choosing interpolation functions in the incompressible or nearly incompressible limit typical in metal forming applications moreover it facilitates the use of low order simplex elements i e p1 p1 reducing the degrees of freedom required for the solution in the incompressible limit when stable elements are necessary linearization of the weak form is derived for implementation into a finite element code numerical experiments with p1 p1 elements show that the method is effective in incompressible conditions and can be advantageous in metal forming analysis

variational and projection methods for the volume constraint in finite deformation elasto plasticity this paper focuses on the treatment of volume constraints which in the context of elasto plasticity typically arise as a result of assuming volume preserving plastic flow projection methods based on the modification of the discrete gradient operator b often proposed on an ad hoc basis are systematically obtained in the variational context furnished by a three field hu washizu principle the fully nonlinear formulation proposed here is based on a local multiplicative split of the deformation gradient into volume preserving and dilatational parts without relying on rate forms of the weak form of momentum balance this approach fits naturally in a formulation of plasticity based on the multiplicative decomposition of the deformation gradient and enables one to exactly enforce the condition of volume preserving plastic flow within the framework proposed in this paper rate forms and incrementally objective algorithms are entirely bypassed

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

the kink of cellular automaton rule 18 performs a random walk we give an exact characterization of the movement of a single kink in the elementary cellular automaton rule 18 it is a random walk with independent increments as well as independent delay times its statistical parameters are computed to confirm the earlier simulation results by grassberger

techniques for stress and strain controlled homogenization of inelastic periodic composites abstract a framework is developed within the context of the displacement finite element method for applying periodic boundary conditions to uni period basic cells under the influence of applied macroscopically homogeneous stresses the resulting approach supplements the more common strain controlled method which is also presented and which relies on an additive decomposition of the displacement field into linear imposed and periodic unknown contributions as implemented here the stress controlled method employs a penalty formulation to insure that the displacement solution satisfies the linearperiodic decomposition the performance of both the stress and strain driven methods are demonstrated and assessed on a complete set of homogenization computations for an elasto plastic composite

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

shakedown and steady state responses of elastic plastic solids in large displacements abstract elastic perfectly plastic solids or structures subjected to loads quasi statically varying within a specified domain are addressed in the framework of large displacements and the additive strain decomposition rule on the ground of drucker s principle of stability in the large an appropriate stability requisite called d stability is formulated as the positive definiteness property of a specific functional sum of the second variation of the helmholtz free energy with an additional term depending on higher order geometry change effects for a d stable structure for which the additive strain decomposition rule is applicable melan s and koiter s theorems of classical shake down theory are reconsidered and reformulated for large displacements the extended melan and koiter theorems so established save the essential features of the classical ones but exhibit a greater formal complexity with consequent difficulties for engineering applications for structures subjected to periodic loads it is shown that as long as the structure does not incur loss of d stability a long term steady state response or steady cycle occurs which exhibits the same periodicity characteristics as in case of small displacements that is the second piola kirchhoff stresses and the plastic green lagrange strain rates become periodic as the load a few illustrative numerical results are presented

on modified displacement version of the non linear theory of thin shells abstract we discuss the non linear theory of thin shells expressed in terms of displacements of the shell reference surface as the only independent field variables the formulation is based on the principle of virtual work postulated for the reference surface in our approach 1 the vector equilibrium equations are represented through components in the deformed contravariant surface base and using the compatibility conditions the resulting tangential equilibrium equations are additionally simplified 2 at the shell boundary the new scalar function of displacement derivatives is defined and new sets of four work conjugate static and geometric boundary conditions are derived as well as 3 for prescribed shell geometry all non linear shell relations are generated automatically by two packages set up in mathematica the displacement boundary value problem and the associated homogeneous shell buckling problem are generated exactly without using any additional approximations following from errors of the constitutive equations both problems are extremely complex and available only in the computer memory such an approach allows us to account also for those a few supposedly small terms which may be critical for finding the correct buckling load of shells sensitive to imperfections this approach is used in the accompanying paper by opoka and pietraszkiewicz opoka s pietraszkiewicz w 2009 on refined analysis of bifurcation buckling for the axially compressed circular cylinder international journal of solids and structures 46 3111 3123 to perform the refined numerical analysis of bifurcation buckling for the axially compressed circular cylinder

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

percolation games probabilistic cellular automata and the hard core model let each site of the square lattice mathbb z 2 be independently assigned one of three states a trap with probability p a target with probability q and open with probability 1 p q where 0 p q 1 consider the following game a token starts at the origin and two players take turns to move where a move consists of moving the token from its current site x to either x 0 1 or x 1 0 a player who moves the token to a trap loses the game immediately while a player who moves the token to a target wins the game immediately is there positive probability that the game is drawn with best play i e that neither player can force a win this is equivalent to the question of ergodicity of a certain family of elementary one dimensional probabilistic cellular automata pca these automata have been studied in the contexts of enumeration of directed lattice animals the golden mean subshift and the hard core model and their ergodicity has been noted as an open problem by several authors we prove that these pca are ergodic and correspondingly that the game on mathbb z 2 has no draws on the other hand we prove that certain analogous games do exhibit draws for suitable parameter values on various directed graphs in higher dimensions including an oriented version of the even sublattice of mathbb z d in all d ge 3 this is proved via a dimension reduction to a hard core lattice gas in dimension d 1 we show that draws occur whenever the corresponding hard core model has multiple gibbs distributions we conjecture that draws occur also on the standard oriented lattice mathbb z d for d ge 3 but here our method encounters a fundamental obstacle

limit distributions of u statistics resampled by symmetric stable laws if y i and v i are independent random sequences such thaty i are i i d random variables belonging to the normal domain of attraction of a symmetric stable law 0 2 andv i are i i d random variables then the limit distributions of u statistics n 1 alpha sum limits 1 leqq i t ldots i d leqq n y i 1 ldots y i d f v i 1 ldots v i d coincide with the probability laws of multiple stochastic integralsx d f f t 1 t d dx t d with respect to a symmetric stable processx t

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

an analysis of a new class of integration algorithms for elastoplastic constitutive relations an accuracy analysis of a new class of integration algorithms for finite deformation elastoplastic constitutive relations recently proposed by the authors is carried out in this paper for simplicity attention is confined to infinitesimal deformations the integration rules under consideration fall within the category of return mapping algorithms and follow in a straightforward manner from the theory of operator splitting applied to elastoplastic constitutive relations general rate independent and rate dependent behaviour with plastic hardening or softening associated or non associated flow rules and nonlinear elastic response can be efficiently treated within the present framework isoerror maps are presented which demonstrate the good accuracy properties of the algorithm even for strain increments much larger than the characteristic strains at yielding

two stress update algorithms for large strains accuracy analysis and numerical implementation two algorithms for the stress update i e time integration of the constitutive equation in large strain solid mechanics are compared from an analytical point of view the order of the truncation error associated to the numerical integration is deduced for each algorithm a priori using standard numerical analysis this accuracy analysis has been performed by means of a convected frame formalism which also allows a unied derivation of both algorithms in spite of their inherent dierences then the two algorithms are adapted from convected frames to a xed cartesian frame and implemented in a small strain nite element code the implementation is validated by means of a set of simple deformation paths simple shear extension extension and compression extension and rotation and two benchmark tests in non linear mechanics the necking of a circular bar and a shell under ring loads in these numerical tests the observed order of convergence is in very good agreement with the theoretical order of convergence thus corroborating the accuracy analysis 1997 john wiley sons ltd

on a p stable multiple integral ii we construct a multiple strictly p stable stochastic integral as a dunford type integral with respect to a l q valued vector measure 1qp 2

on the numerical implementation of inelastic time dependent and time independent finite strain constitutive equations in structural mechanics abstract a number of complex issues are resolved in a way that allows the incorporation of finite strain inelastic material behavior into the piecewise numerical construction of solutions in solid mechanics without recourse to extensive continuum mechanics preliminaries an elementary time independent plasticity model an elementary time dependent creep model and a viscoelastic model are introduced as examples of constitutive equations which are routinely used in engineering calculations the constitutive equations are all suitable for problems involving large deformations and finite strains the plasticity and creep models are in rate form and use the symmetric part of the velocity gradient or the stretching to compute the co rotational time derivative of the cauchy stress the viscoelastic model computes the current value of the cauchy stress from a hereditary integral of a materially invariant form of the stretching history the current configuration is selected for evaluation of equilibrium as opposed to either the reference configuration or the last established equilibrium configuration the process of strain incrementation is examined in some depth the stretching evaluated at the mid interval and multiplied by the time step is identified as the appropriate finite strain increment to use with the selected forms of the constitutive equations discussed is the conversion of rotation rates based on the spin into incremental orthogonal rotations these rotations are used to update stresses and state variables due to rigid body rotation during the load increment comments and references to the literature are directed at numerical integration of the constitutive equations with an emphasis on doing this accurately if not exactly for any time step and stretching this material taken collectively provides an approach to numerical implementation which is marked by its simplicity

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

consistent tangent operators for rate independent elastoplasticity abstract it is shown that for problems involving rate constitutive equations such as rate independent elastoplasticity the notion of consistency between the tangent stiffness operator and the integration algorithm employed in the solution of the incremental problem plays a crucial role in preserving the quadratic rate of asymptotic convergence of iterative solution schemes based upon newton s method within the framework of closest point projection algorithms a methodology is presented whereby tangent operators consistent with this class of algorithms may be systematically developed to wit associative j 2 flow rules with general nonlinear kinematic and isotropic hardening rules as well as a class of non associative flow rules are considered the resulting iterative solution scheme preserves the asymptotic quadratic convergence characteristic of newton s method whereas use of the socalled elastoplastic tangent in conjunction with a radial return integration algorithm a procedure often employed results in newton type of algorithms with suboptimal rate of convergence application is made to a set of numerical examples which include saturation hardening laws of exponential type

on the multiple stable integral we prove that the n fold stochastic integral mathop smallint limits left 0 1 right n f left t 1 t n right dz left t 1 right dz left t n right with respect to a p stable levy process z t t 0 1 p 1 2 exists as a continuous linear mapping from the zygmund space l p log l n 1 of functions f 0 1 n to l q for any q p this strengthens the author s earlier result 15 the proof is based on an interpolation theorem due to bennett and rudnick 2

percolation and disorder resistance in cellular automata we rigorously prove a form of disorder resistance for a class of one dimensional cellular automaton rules including some that arise as boundary dynamics of two dimensional solidification rules specifically when started from a random initial seed on an interval of length l with probability tending to one as l the evolution is a replicator that is a region of space time of density one is filled with a spatially and temporally periodic pattern punctuated by a finite set of other finite patterns repeated at a fractal set of locations on the other hand the same rules exhibit provably more complex evolution from some seeds while from other seeds their behavior is apparently chaotic a principal tool is a new variant of percolation theory in the context of additive cellular automata from random initial states

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

some computational aspects of large deformation rate dependent plasticity problems abstract the governing equations for a finite element formulation of boundary value problems for large deformation metal forming processes are derived using a principle of virtual work formulated in a lagrangian reference system the updated lagrangian method is used to simplify the equations the resulting nonlinear equilibrium equations are solved using newton s method a constitutive model for large deformation rate dependent plasticity is developed the model incorporates a one parameter implicit integration operator for stability and accuracy the stressrate strain rate relation is written in terms of the jaumann rate of stress numerous example problems are solved to demonstrate the effectiveness of the numerical algorithms

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

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 return mapping algorithm for plane stress elastoplasticity an unconditionally stable algorithm for plane stress elastoplasticity is developed based upon the notion of elastic predictor return mapping plastic corrector enforcement of the consistency condition is shown to reduce to the solution of a simple nonlinear equation consistent elastoplastic tangent moduli are obtained by exact linearization of the algorithm use of these moduli is essential in order to preserve the asymptotic rate of quadratic convergence of newton methods the accuracy of the algorithm is assessed by means of iso error maps the excellent performance of the algorithm for large time steps is illustrated in numerical experiments

stochastic integrals a combinatorial approach a combinatorial definition of multiple stochastic integrals is given in the setting of random measures it is shown that some properties of such stochastic integrals formerly known to hold in special cases are instances of combinatorial identities on the lattice of partitions of a set the notion of stochastic sequences of binomial type is introduced as a generalization of special polynomial sequences occuring in stochastic integration such as hermite poisson charlier and kravchuk polynomials it is shown that identities for such polynomial sets have a common origin

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

the one dimensional exactly 1 cellular automaton replication periodicity and chaos from finite seeds in the exactly 1 cellular automaton also known as rule 22 every site of the one dimensional lattice is either in state 0 or in state 1 and a synchronous update rule dictates that a site is in state 1 next time if and only if it sees a single 1 in its three site neighborhood at the current time we analyze this rule started from finite seeds i e those initial configurations that have only finitely many 1 s three qualitatively different types of evolution are observed replication periodicity and chaos we focus on rigorous results assisted by algorithmic searches for the first two behaviors in particular we explain why replication is observed so frequently and present a method for collecting the smallest periodic seeds some empirical observations about chaotic seeds are also presented

random multilinear forms on etudie la convergence des formes multilineaires f n n k x n 1 x n k dans des variables aleatoires independantes symetriques

accuracy and stability of integration algorithms for elastoplastic constitutive relations an analysis of accuracy and stability of algorithms for the integration of elastoplastic constitutive relations is carried out in this paper reference is made to a very general internal variable formulation of plasticity and to two families of algorithms that generalize the well known trapezoidal and midpoint rules to fit the present context other integration schemes such as the radial return mean normal and closest point procedures are particular cases of this general formulation the meaning of first and second order accuracy in the presence of the plastic consistency condition is examined in detail and the criteria derived are used to identify two second order accurate members of the proposed algorithms a general methodology is also derived whereby the numerical stability properties of integration schemes can be systematically assessed with the aid of this methodology the generalized midpoint rule is seen to have far better stability properties than the generalized trapezoidal rule finally numerical examples are presented that illustrate the performance of the algorithms

design sensitivity analysis in large deformation elasto plastic and elasto viscoplastic problems implicit time integration algorithm derived by simo for his large deformation elasto plastic constitutive model is generalized for the case of isotropy and associative flow rule towards viscoplastic material behaviour and consistently differentiated with respect to its input parameters combining it with the general formulation of design sensitivity analysis dsa for non linear finite element transient equilibrium problem we come at a numerically efficient closed form finite element formulation of dsa for large deformation elasto plastic and elasto viscoplastic problems with various types of design variables material constants shape parameters the paper handles several specific issues like the use of a non algorithmic coefficient matrix or sensitivity discontinuities at points of instantaneous structural stiffness change computational examples demonstrate abilities of the formulation and quality of results copyright 2005 john wiley sons ltd

deformation analysis for finite elastic plastic strains in a lagrangean type description abstract a general concept is presented to analyse the deformation of structures undergoing arbitrarily large elastic and arbitrarily large plastic strains based on the multiplicative decomposition of the deformation gradient into elastic and plastic contributions the kinematics of two superposed finite non coaxial deformations are investigated lagrangean type elastic and plastic stretch tensors are introduced and multiplicative decompositions of the total stretch into these elastic and plastic stretches are derived it is shown that the result is independent of any decomposition of the total rotation into an elastic and a plastic rotation for the second superposed deformation the total lagrangean logarithmic hencky strain tensor with corresponding elastic and plastic logarithmic strains is defined if in a large deformation analysis the first deformation is updated such that the second deformation is constrained to be moderately large then the total logarithmic strain tensor of the second deformation can be additively decomposed into purely elastic and purely plastic parts this enables an appropriate formulation of constitutive equations for isotropic hyperelastic material behavior with associated flow rule and evolution laws for combined isotropic kinematic hardening work conjugate to the elastic logarithmic strain tensor is a back rotated kirchhoff stress tensor the rotational change of its reference configuration during the update is given explicitly finally the principle of virtual work with corresponding equilibrium equations and static and geometric boundary conditions is given the virtual work functional is transformed to deliver the consistent tangent stiffness matrix as basis for a finite element solution algorithm

contraction and decoupling inequalities for multilinear forms and u statistics we prove decoupling inequalities for random polynomials in independent random variables with coefficients in vector space we use various means of comparison including rearrangement invariant norms e g orlicz and lorentz norms tail distributions tightness hypercontractivity and so forth

theory and numerical analysis of shells undergoing large elastic strains abstract a non linear bending theory of rubber like shells undergoing large elastic strains is proposed the theory is based on a relaxed normality hypothesis and the incompressibility condition using series expansion in the normal direction and applying some estimation technique a consistent first approximation and a simplest approximation to the elastic strain energy of the shell are constructed lagrangian displacement shell equations are derived and the incremental shell deformation is considered the numerical results presented for one and two dimensional large strain shell problems confirm the accuracy and efficiency of the proposed shell theory

a field boundary element formulation for material nonlinear problems at finite strains abstract a boundary element formulation for geometrical nonlinear problems is presented which is applicable for arbitrary constitutive equations based on the concept of hyperelastic response relative to an intermediate configuration this includes purely hyperelastic problems at large elastic strains in contrast to other formulations the total lagrange framework employed in this work significantly increases the numerical efficiency as the system matrices remain constant during the incremental solution and have to be computed only once a nonlinear set of equations with an identical structure for both boundary and internal unknowns is derived the basic unknowns are displacement gradients which are calculated via integral equations and not by differentiation of the shape functions the formulation can be consistently linearised to allow for the implementation of efficient iteration schemes using gradients e g the newton raphson scheme

applications of ergodic theory and sofic systems to cellular automata abstract we indicate a mathematical framework for analysing the evolution of cellular automata those automata obeying an additive rule are shown to be the same as endomorphisms of a compact abelian group and therefore their statistical and dynamical behavior can be told exactly by using fourier analysis and ergodic theory those obeying certain nonlinear rules are closely tied to a finitely described object called a sofic system but the underlying statistics appear to be more complicated and interesting we conclude by formulating several conjectures about one such system

the sharp threshold for bootstrap percolation in all dimensions in r neighbour bootstrap percolation on a graph g a typically random set a of initially infected vertices spreads by infecting at each time step vertices with at least r already infected neighbours this process may be viewed as a monotone version of the glauber dynamics of the ising model and has been extensively studied on the d dimensional grid n d the elements of the set a are usually chosen independently with some density p and the main question is to determine p c n d r the density at which percolation infection of the entire vertex set becomes likely in this paper we prove for every pair d ge r ge 2 that there is a constant l d r such that p c n d r l d r o 1 log r 1 n d r 1 as n o infty where log r denotes an r times iterated logarithm we thus prove the existence of a sharp threshold for percolation in any fixed number of dimensions moreover we determine l d r for every pair d r

objectivity of incremental constitutive relations over finite time steps in computational finite deformation analyses abstract computational techniques which preserve the objectivity of incremental constitutive relations for finite elastic or inelastic materials for finite time steps during which the material elements may undergo finite rotations are presented these techniques are useful in finite deformation finite rotations and finite strain computational analysis of two as well as three dimensional solids and structures

robust periodic solutions and evolution from seeds in one dimensional edge cellular automata we study one dimensional cellular automata ca with values 0 and 1 we assume that such ca are started from semi infinite configurations those that have 0 s to the left of some site and we focus on the identification of robust periodic solutions rps which when observed from the left edge of the light cone advance into any environment with positive velocity we then utilize rps and related concepts to analyze ca dynamics from seeds i e initial configurations with finitely many 1 s

quadrilateral elements for the solution of elasto plastic finite strain problems in this paper two plane strain quadrilateral elements with two and four variables are proposed these elements are applied to the analysis of finite strain elasto plastic problems the elements are based on the enhanced strain and b bar methodologies and possess a stabilizing term the pressure and dilatation fields are assumed to be constant in each element s domain and the deformation gradient is enriched with additional variables as in the enhanced strain methodology the formulation is deduced from a four field functional based on the imposition of two constraints annulment of the enhanced part of the deformation gradient and the relation between the assumed dilatation and the deformation gradient determinant the discretized form of equilibrium is presented and the analytical linearization is deduced to ensure the asymptotically quadratic rate of convergence in the newton raphson method the proposed formulation for the enhanced terms is carried out in the isoparametric domain and does not need the usually adopted procedure of evaluating the jacobian matrix in the centre of the element the elements are very effective for the particular class of problems analysed and do not present any locking or instability tendencies as illustrated by various representative examples copyright 2001 john wiley sons ltd

sharp metastability threshold for two dimensional bootstrap percolation in the bootstrap percolation model sites in an l by l square are initially independently declared active with probability p at each time step an inactive site becomes active if at least two of its four neighbours are active we study the behaviour as p 0 and l simultaneously of the probability i l p that the entire square is eventually active we prove that i l p 1 if and i l p 0 if where 2 18 we prove the same behaviour with the same threshold for the probability j l p that a site is active by time l in the process on the infinite lattice the same results hold for the so called modified bootstrap percolation model but with threshold 2 6 the existence of the thresholds settles a conjecture of aizenman and lebowitz 3 while the determination of their values corrects numerical predictions of adler stauffer and aharony 2

algebraic properties of cellular automata cellular automata are discrete dynamical systems of simple construction but complex and varied behaviour algebraic techniques are used to give an extensive analysis of the global properties of a class of finite cellular automata the complete structure of state transition diagrams is derived in terms of algebraic and number theoretical quantities the systems are usually irreversible and are found to evolve through transients to attractors consisting of cycles sometimes containing a large number of configurations

double stochastic integrals random quadratic forms and random series in orlicz spaces let x t t geq 0 be a process with independent symmetric and stationary increments and let xi i be i i d symmetric real random variables we provide a characterization of functions f s t s t geq 0 such that the double integral int int f s t dx s dx t exists a characterization of infinite matrices alpha ij such that the double series sum alpha ij xi i xi j converges a s and a characterization of orlicz space l psi valued sequences a i for which the series sum a i xi i converges a s in l psi the above three problems are closely related

dependent percolation in two dimensions for a natural number k define an oriented site percolation on 2 as follows let xi yj be independent random variables with values uniformly distributed in 1 k declare a site i j 2closed if xi yj and open otherwise peter winkler conjectured some years ago that if k 4 then with positive probability there is an infinite oriented path starting at the origin all of whose sites are open i e there is an infinite path p i0 j0 i1 j1 such that 0 i0 i1 0 j0 j1 and each site in jn is open rather surprisingly this conjecture is still open in fact it is not known whether the conjecture holds for any value of k in this note we shall prove the weaker result that the corresponding assertion holds in the unoriented case if k 4 then the probability that there is an infinite path that starts at the origin and consists only of open sites is positive furthermore we shall show that our method can be applied to a wide variety of distributions of xi and yj independently peter winkler 14 has recently proved a variety of similar assertions by different methods

adaptive finite element methods in computational mechanics we present a general approach to adaptivity for finite element methods and give applications to linear elasticity non linear elasto plasticity and nonlinear conservation laws including numerical results

replication in one dimensional cellular automata abstract in a cellular automaton ca replication is the ability to indefinitely generate copies of a finite collection of patterns starting from finite seeds a transparent feature of additive ca replication mechanisms are less clear in the absence of additivity this paper investigates such dynamics through several examples for the 1 or 2 rule and its generalizations replication is inevitable and we investigate self organization properties in the perturbed exactly 1 rule we study frequency of replicators and the new phenomenon is called quasireplication the last ca is the extended 1 or 3 rule which allows for replication on different backgrounds we employ a mixture of rigorous and empirical techniques

comparison of moments for tangent sequences of random variables it is shown that for all tangent sequences dn and en of nonnegative or conditionally symmetric random variables and for every function satisfying the growth condition 2x x the following inequality holds e phi left mathop sup limits n left sum limits k 1 n d k right right leqq ce phi left mathop sup limits n left sum limits k 1 n e k right right this generalizes results of j zinn and proves a conjecture of s kwapien and w a woyczynski

newton like methods with limited storage for the solution of elasto viscoplasticity problems quasi newton and secant newton methods have been successful in solving problems of elasto plasticity in this paper these methods are applied to elasto viscoplastic analyses in an algorithm coupling the global solution for displacement increments with local newton raphson stress iteration at each sampling point the familiar rank two dfp and bfgs methods are generalized to the broyden family of algorithms and this generalization is also used to produce a secant newton algorithm corresponding to hoshino s method consideration is given to the most effective strategy when only a limited number of quasi newton iterates may be stored overwriting re start or continuation in secant newton form results are presented for a tunnelling problem using both plane strain and axisymmetric formulations

plastic potential theory in large strain elastoplasticity abstract the classical theory of isotropie rate independent strain hardening elastoplasticity based on the invariance of elastic properties is reformulated as an incremental state variable theory in this eulerian spatial formulation state is described in terms of a scalar work hardening parameter and a single finite strain elastic deformation tensor that fixes the elastic stretch ellipsoid in the current configuration this theoretically efficient formulation proves amenable to the consideration of material stability as the postulate of il iushin is shown to rigorously extend the classical plastic potential theory into the larger strain deformation regime in virtually identical form special care is taken to establish the precise connection between these theoretical forms and those of lee as well as the more general forms of green and naghdi small strain linearization then serves to justify a particularly convenient frame invariant generalization of the classical prandtl reuss equations

multiple integration with respect to poisson and levy processes necessary and sufficient conditions are given for the existence of a multiple stochastic integral of the form fdx1 dxd where x1 xd are components of a positive or symmetric pure jump type levy process in d conditions are also given for a sequence of integrals of this type to converge in probability to zero or infinity or to be tight all arguments proceed via reduction to the special case of poisson integrals

numerical integration of rate constitutive equations in finite deformation analysis abstract in analysis of finite deformation problems the use of constitutive equations in rate form is often required in a spatial setting these equations may express a relationship between some objective rate of spatial stress tensor and the rate of deformation constitutive equations of this type characterize a variety of material models including hyperelasticity hypoelasticity and elastoplasticity employing geometrical concepts a family of unconditionally stable and incrementally objective algorithms is proposed for the integration of such rate constitutive equations these algorithms which are appropriate for finite deformation analysis are applicable to any choice of stress rate and in most cases employ quantities that arise naturally in the context of finite element analysis examples illustrate the objectivity and accuracy of the algorithms

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

on the numerical integration of a class of pressure dependent plasticity models an unconditionally stable algorithm for the numerical integration of elastoplastic pressure dependent constitutive relations is analysed in detail in this paper the application of the method to plane stress problems in which the out of plane strain component is not defined kinematically is discussed the tangent moduli resulting from this integration algorithm are obtained by consistent linearization of the elastoplastic constitutive equations the algorithm is applied to gurson s constitutive model some one dimensional problems are solved and comparisons with exact solutions are made the paper closes with a numerical study of the necking of an axi symmetric specimen using gurson s plasticity model to describe the constitutive behaviour of the material

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