strain and stress based continuum damage models ii computational aspects abstract in part i of this work we developed continuum isotropic and anisotropic elastoplastic damage models formulated either in strain space on the basis of the effective stress concept or in stress space and employing the dual notion of effective strain in part ii we consider in detail the variational formulation and subsequent numerical implementation of these models the former relies crucially on the notions of maximum plastic and maximum damage dissipation the latter makes systematic use of the operator splitting methodology to derive unconditionally stable algorithms for the numerical integration of the elastoplastic damage equations of evolution appropriate extensions to treat the proposed rate dependent viscous damage are also presented for stress based damage models our numerical treatment relies on a new three step operator split the algorithms developed lead to simple and efficient stress update procedures suitable for large scale finite element calculations application is made to a class of inviscid and rate dependent cap models with an isotropic strain based damage mechanism remarkably good agreement with existing experimental data that includes complicated stress paths is obtained numerical examples are also presented that demonstrate the good performance of the proposed algorithms

coupled multi scale cohesive modeling of failure in heterogeneous adhesives a multi scale cohesive numerical framework is proposed to simulate the failure of heterogeneous adhesively bonded systems this multi scale scheme is based on hill s variational principle of energy equivalence between the higher and lower level scales it provides an easy way to obtain accurate homogenized macroscopic properties while capturing the physics of failure processes at the micro scale in sufficient detail we use an isotropic rate dependent damage model to mimic the failure response of the constituents of heterogeneous adhesives the finite element method is used to solve the equilibrium equation at each scale a nested iterative scheme inspired by the return mapping algorithm used in computational inelasticity is implemented we propose a computationally attractive technique to couple the macro and micro scales for rate dependent constitutive laws we introduce an adhesive patch test to study the numerical performance including spatial and temporal convergence of the multi scale scheme we compare the solution of the multi scale cohesive scheme with a direct numerical simulation finally we solve mode i and mode ii fracture problems to demonstrate failure at the macro scale

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

elastoplastic damage modelling including the gradient of damage formulation and computational aspects a framework for continuum elastoplastic damage modelling which employs irreversible thermodynamics and internal state variables is investigated the damage part of the modelling involves the gradient of damage quantity which together with the equations of motion are issued from a new formulation of the principle of virtual power recently proposed it is shown how the plastic part of the modelling takes its standard format next we consider in detail the variational formulation and subsequent numerical implementation of the elastoplastic damage models the development of an algorithm consistent with the present formulation is given where for the plastic part it leaves the standard return mapping algorithms unchanged application is made to different classical rate independent and rate dependent plastic models with the presence of a damage mechanism the numerical implementation in the context of the finite element method is discussed in detail and a set of representative applications and numerical examples is given

discrete breathers in honeycomb fermi pasta ulam lattices we consider the two dimensional fermi pasta ulam lattice with hexagonal honeycomb symmetry which is a hamiltonian system describing the evolution of a scalar valued quantity subject to nearest neighbour interactions using multiple scale analysis we reduce the governing lattice equations to a nonlinear schrodinger nls equation coupled to a second equation for an accompanying slow mode two cases in which the latter equation can be solved and so the system decoupled are considered in more detail firstly in the case of a symmetric potential we derive the form of moving breathers we find an ellipticity criterion for the wavenumbers of the carrier wave together with asymptotic estimates for the breather energy the minimum energy threshold depends on the wavenumber of the breather we find that this threshold is locally maximised by stationary breathers secondly for an asymmetric potential we find stationary breathers which even with a quadratic nonlinearity generate no second harmonic component in the breather plots of all our findings show clear hexagonal symmetry as we would expect from our lattice structure finally we compare the properties of stationary breathers in the square triangular and honeycomb lattices

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

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

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

stationary breather modes of generalized nonlinear klein gordon lattices this paper studies the form of stationary breather modes in discrete generalized nonlinear klein gordon equations with symmetric and non symmetric potential energy functions in the case of static breathers the discrete nature of the spatial dimension has a much more subtle effect on the breather than the moving breather mode this effect is analysed using a variety of approximating partial differential equations whose solutions are found by using an extended multiple scales asymptotic approach to reduce the equation to a nonlinear schrodinger equation at leading order and more complex equations at higher order where secularity conditions are required to fully specify the solution as well as the much studied discrete sine gordon equation the methods are demonstrated on a discrete nonlinear klein gordon equation with second neighbour interactions and non symmetric on site potential new partial differential equations which approximate these lattice systems are also proposed and analysed

variational approximations to breather modes in the discrete sine gordon equation abstract long lived breather like modes have been observed in the numerical simulations of discrete sine gordon systems however little analytical work has been carried out on these systems we show how to use the variational form of nonlinear klein gordon equations to find corrections to breather modes caused by discreteness

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

elastoplastic material with isotropic damage the two damage parameters needed to describe isotropic damage are shown to define a damage domain and the evolution of damage is defined by a path within that domain physical limitations on the path to full damage are developed a thermodynamically consistent set of constitutive equations with which the damage path can be evaluated is developed for a specific material

the use of displacement discontinuities in a rate dependent medium abstract this paper illustrates the use of displacement discontinuities in a rate dependent elastoplastic damage model rate dependency is considered in the framework of perzyna viscoplasticity displacement discontinuities are incorporated using the partition of unity property of finite element shape functions a combined continuous discontinuous approach is proposed in which a discontinuity is only inserted after a certain amount of damage various examples illustrate the applicability of the combined approach to granular and quasi brittle materials with and without fibre reinforcement

on energy based coupled elastoplastic damage theories constitutive modeling and computational aspects abstract novel energy based coupled elastoplastic damage theories are presented in this paper the proposed formulation employs irreversible thermodynamics and internal state variable theory for ductile and brittle materials at variance with lemaitre s work on damage elastoplasticity the present formulation renders rational thermodynamic potential and damage energy release rate in contrast to previous work by simo and ju featuring an additive split of the stress tensor current formulation assumes an additive split of the strain tensor it is shown that the strain split damage elastoplasticity formulation leads to more robust tangent moduli than the stress split formulation the plastic flow rule and hardening law are characterized in terms of the effective quantities viz the effective stress space plasticity this mechanism is both physically well motivated and computationally efficient further a fourth order anisotropic damage mechanism is proposed for brittle materials rational mechanisms are also presented to account for the microcrack opening and closing operations as well as the strain rate dependency of microcrack growth efficient computational algorithms for proposed elasloplaslic damage models are subsequently explored by making use of the operator splitting methodology in particular new three step operator split algorithms are presented application is made to a class of inviseid and rate dependent cap damage models for concrete and mortar experimental validations are also given to illustrate the applicability of the proposed damage models

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

a displacement like finite element model for j2 elastoplasticity variational formulation and finite step solution abstract the displacement like finite element formulation for finite step j2 elastoplasticity is revisited in this paper the classical computational strategy according to which plastic loading is tested at the gauss points of each element and an independent return mapping algorithm is performed for given incremental displacements is consistently derived from a suitably discretized version of a min max variational principle the sequence of solution phases to be performed within each load step adopting a full newton s method is illustrated in detail and the importance of a correct update of the plastic strains is emphasized it is further shown that in order to increase the rate of convergence and the stability properties of the newton s method the consistent elastoplastic tangent operator must be exploited even at the first iteration of each load step subsequent to the first yielding of the structural model this is in contrast with the traditional implementation according to which the elastic operator is used at the first iteration of each load step the effectiveness of the present approach is shown by a set of numerical examples referred to plane strain problems

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

free c actions on c3 are translations let x be a smooth contractible three dimensional affine algebraic variety with a free algebraic c action on it such that s x c is smooth we prove that x is isomorphic to s c and the action is induced by a translation on the second factor as a consequence we show that any free algebraic c action on c 3 is a translation in a suitable coordinate system

on the compactifications of contractible affine threefolds and the zariski cancellation problem in this article we consider the compactifications of some kinds of contractible smooth affine threefolds and the characterization of the affine 3 space gif13 keeping the 3 dimensional zariski cancellation problem in mind we classify the compactifications of contractible smooth affine threefolds with a certain condition concerning the numerical property of boundary divisors with respect to compactifications and then we show that if an affine threefold x satisfies x gif11 gif14 and this numerical condition then x is isomorphic to the affine 3 space gif13

asymptotic analysis of combined breather kink modes in a fermi pasta ulam chain we find approximations to travelling breather solutions of the one dimensional fermi pasta ulam fpu lattice both bright breather and dark breather solutions are found we find that the existence of localised bright solutions depends upon the coefficients of cubic and quartic terms of the potential energy generalising an earlier inequality derived by james cr acad sci paris 332 581 2001 we use the method of multiple scales to reduce the equations of motion for the lattice to a nonlinear schr o dinger equation at leading order and hence construct an asymptotic form for the breather we show that in the absence of a cubic potential energy term the lattice supports combined breathing kink waveforms the amplitude of breathing kinks can be arbitrarily small as opposed to traditional monotone kinks which have a nonzero minimum amplitude in such systems we also present numerical simulations of the lattice verifying the shape and velocity of the travelling waveforms and confirming the long lived nature of all such modes

numerical analysis of anisotropic ductile continuum damage abstract the paper deals with the numerical analysis of large elastic plastic deformation behavior of anisotropically damaged ductile solids based on a generalized macroscopic theory within the framework of nonlinear continuum damage mechanics estimates of the stress and strain histories are obtained from a straightforward numerical integration algorithm based on operator split methodology which employs an inelastic damage plastic predictor followed by an elastic corrector step the finite element method is used to approximate the linearized variational problem furthermore identification of material parameters is discussed numerical simulation of the elastic plastic deformation behavior of damaged tension specimens demonstrate the efficiency of the formulation

quasi continuum approximations to lattice equations arising from the discrete nonlinear telegraph equation we show how quasi continuum methods can be used to construct approximate solutions of nonlinear differential delay equations derived from symmetry reductions of the discrete nonlinear telegraph equation travelling wave solutions are proven to exist and the existence of solutions to two other symmetry reductions are studied two of the less familiar reductions are studied the first supports both a one parameter family of single pulse solitary wave type solutions and a one parameter family of periodic waves the size and shape of these waves are examined using the quasi continuum technique this approximates the differential difference equation with a higher order differential equation which is integrated and analysed using phase plane techniques in the large amplitude limit the shape of the pulse approaches a limiting form which has a corner at its peak the manner of this approach is elucidated using matched asymptotic expansions the second reduction though differing only by the addition of a single term appears not to support the solitary wave type of solution even in the limit where the additional term is premultiplied by an asymptotically small constant

the equivalence of stress and strain based plasticity theories abstract strain space plasticity theory has been advocated by a number of researchers as a viable alternative to its stress space counterpart however there appears to be a certain level of uncertainty about the equivalence of stress and strain based plasticity theories this paper attempts to clarify some issues raised by casey and naghdi 2 concerning the equivalence of these two theories by providing the alternative conjugate expressions for the loading criteria it will be shown that the two formulations are indeed equivalent in substance and produce equivalent expressions for the plastic strain rate provided that the material laws used are identical in both approaches the use of the strain space formulation in many cases for example when dealing with strain softening materials is deemed to be convenient and therefore desirable nevertheless it is not essential the findings in this article exemplify the statement made by drucker 4 the use of a strain space or a stress space is equally permissible but may not be equally convenient for one purpose or the other

a tangent secant approach to rate independent elastoplasticity formulations and computational issues a general and robust solution procedure for nonlinear finite element equations in small strain elastoplastic structural problems is presented its peculiar feature lies in the choice of the most suitable constitutive operator to be adopted at each iteration of a generic load step in order to ensure the utmost stability and convergence rate namely the consistent tangent operator is replaced by a secant one or vice versa whether the adopted norm of the residual does not or does conveniently decrease at the current iteration the secant operator is defined as to recover the finite step increment of the plastically admissible stress from the total not iterative strain increment the original formulation of the solution procedure consisting of alternate tangent and secant iterations is then extended to achieve an effective coupling with line searches the excellent performances of the two procedures are illustrated by numerical examples carried out for typical benchmark problems in plane strain and three dimensional cases

solitary waves with damped oscillatory tails an analysis of the fifth order korteweg de vries equation abstract we construct oscillatory solitary wave solutions of a fifth order korteweg de vries equation where the oscillations decay at infinity these waves arise as a bifurcation from the linear dispersion curve at that wavenumber where the linear phase speed and group velocity coincide our approach is a wave packet analysis about this wavenumber which leads in the first instance to a higher order nonlinear schrodinger equation from which we then obtain the steady solitary wave solution we then describe a complementary normal form analysis which leads to the same result in addition we derive the nonlinear schrodinger equation for all wavenumbers and list all the various anomalous cases

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

a thermodynamically consistent nonlocal damage model for concrete materials with unilateral effects abstract this paper proposes a thermodynamically consistent constitutive model for concrete materials to solve several problems encountered in the existing damage modeling two separated equivalent strains and the corresponding new damage evolution laws are used to enhance the representation of asymmetrical behaviors of concrete of which the different strengths and post peak behaviors between tension and compression cannot simultaneously be reproduced by the classical strain driven models the crack opening closure criterion related to the unilateral effect is described using the trace of the strain tensor rather than the spectral decomposition of a specific second order tensor which leads to either a discontinuous stress strain response or spurious energy dissipation upon closed load cycles an integral type nonlocal regularization is incorporated to solve the problem of strain softening induced mesh non objectivity involving mesh size and mesh bias dependencies the strain driven formalism and the explicit integration of constitutive equations improve the robustness of the numerical algorithm correlative studies using available benchmark test results are presented to demonstrate the performance of the model

a two field finite element formulation for elasticity coupled to damage a two field variational formulation is proposed for continuum damage mechanics problems this formulation is applied to the numerical solution via the finite element method of initial boundary value problems arising in elasticity coupled to damage the nodal variables are the displacement and damage fields simple examples are presented to demonstrate the advantages and limitations of this formulation

coupled creep elastoplastic damage analysis for isotropic and anisotropic nonlinear materials abstract this paper deals with the computational aspects of a coupled creep elastoplastic damage analysis for anisotropic and as a special case isotropic nonlinear materials a three phase backward euler integration algorithm for stress update is proposed for anisotropic nonlinear materials a general direct stress return mapping algorithm utilising newton raphson iteration is derived the stress vector and scalar variables quantifying the incremental creep plasticity and damage are updated simultaneously for isotropic materials the elasto plastic stress update algorithm for plane stress by jetteur 1986 engng camp 3 251 253 is extended to include creep and damage in addition a simple stress algorithm for the general three dimensional isotropic case is also presented the resulting algorithms are suitable for inclusion in general structural analysis codes the consistent tangent matrix is also formulated for use in a global newton iterative procedure in which structural displacements are sought as the problem unknowns examples are given using the general purpose code lusas in which the algorithms have been implemented

quasi continuum approximations for travelling kinks in diatomic lattices we extend the quasi continuum method of approximation to the case of a diatomic lattice this is illustrated by a lattice in which both small and large atoms interact with first and second nearest neighbours we show that highly accurate quasi continuum techniques may be generalized to determine the shape of nontopological kinks many previous analyses of such systems have found travelling wave solutions only for one particular speed using a second order continuum theory we show i how wave shapes can be found for arbitrary speeds and ii how solution profiles can be calculated from fourth order partial differential equations which approximate the lattice alongside the theoretical analysis we also present numerical simulations of the lattice which demonstrate propagation of the predicted waves through the lattice we show that the particular speed for which solutions have been found in previous studies is a special speed for waves in the lattice but waves can travel for long periods of time at faster or slower speeds whilst slowly relaxing to this critical speed

approximations to solitary waves on lattices iii the monatomic lattice with second neighbour interactions we find new behaviour in a lattice with second neighbour interactions first we find solitary waves with oscillatory spatial decay this is something that cannot occur in lattices with nearest neighbour interactions alone this is found by standard asymptotic analysis which leads to an exact curve in parameter space where this behaviour starts a number of highly accurate quasi continuum approximations are derived and solved one of these suggests a possible method for subsonic solitary waves to cease existing an alternative method of approximation elegantly reveals the differences in shape between subsonic and supersonic solitary waves this is based on the weak form of the differential delay equation for the travelling wave and is derived using the calculus of variations

a plastic nonlocal damage model abstract in the present paper a plastic nonlocal damage model is proposed for studying the mechanical response of structural elements made of cementitious materials a new isotropic damage model which is able to describe the behavior of a wide class of cementitious materials is presented a regularization technique based on the introduction of the damage laplacian in the damage limit function is adopted to overcome the analytical and computational problems induced by the softening constitutive law a drucker prager type of plastic limit function is proposed considering isotropic hardening a numerical procedure based on an implicit backward euler technique for the time integration of the plastic and damage evolution equations is presented to solve each nonlinear step a predictor corrector iterative method is developed within the splitting method in particular the damage evolution is determined solving a constrained minimization problem of a convex functional the proposed algorithm is implemented in a finite element code and it is used to study the structural behavior of elements made of masonry materials

generalized midpoint finite element dynamic analysis of elastoplastic systems the application of a four parameter generalized midpoint integration method to small displacement elastoplastic dynamic problems is considered the integration of the governing relations over a time step is formulated as a non linear algebraic problem finite step problem by making use of generalized variables finite element modelling uniqueness and an extremum characterization of the solution of the finite step problem are discussed sufficient conditions for convergence of various iterative algorithms including newton raphson are given the b stability of the proposed procedure is assessed

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

failure oriented multi scale variational formulation micro structures with nucleation and evolution of softening bands abstract this contribution presents the theoretical foundations of a failure oriented multi scale variational formulation fomf for modeling heterogeneous softening based materials undergoing strain localization phenomena the multi scale model considers two coupled mechanical problems at different physical length scales denoted as macro and micro scales respectively every point at the macro scale is linked to a representative volume element rve and its constitutive response emerges from a consistent homogenization of the micro mechanical problem at the macroscopic level the initially continuum medium admits the nucleation and evolution of cohesive cracks due to progressive strain localization phenomena taking place at the microscopic level and caused by shear bands damage or any other possible failure mechanism a cohesive crack is introduced in the macro model once a specific macroscopic failure criterion is fulfilled the novelty of the present failure oriented multi scale formulation is based on a proper kinematical information transference from the macro to micro scales during the complete loading history even in those points where macro cracks evolve in fact the proposed fomf includes two multi scale sub models consistently coupled i a classical multi scale model clamm valid for the stable macro scale constitutive response ii a novel cohesive multi scale model cohmm valid once a macro discontinuity surface is nucleated for modeling the macro crack evolution when a macro crack is activated two important kinematical assumptions are introduced i a change in the rule that defines how the increments of generalized macro strains are inserted into the micro scale and ii the kinematical admissibility concept from where proper strain homogenization procedures are obtained then as a consequence of the hill mandel variational principle and the proposed kinematical assumptions the fomf provides an adequate homogenization formula for the stresses in the continuum part of the body as well as for the traction acting on the macro discontinuity surface the assumed macro to micro mechanism of kinematical coupling defines a specific admissible rve displacement space which is obtained by incorporating additional boundary conditions non standard boundary conditions nsbc in the new model a consequence of introducing these non standard boundary conditions is that they guarantee the existence of a physically admissible rve size a concept that we call through the paper objectivity of the homogenized constitutive response several numerical examples are presented showing the objectivity of the formulation as well as the capabilities of the new multi scale approach to model material failure problems

damage mechanics basic variables in continuum theories it is shown that any type of material damage that causes stiffness degradation in general requires description by an eighth order damage tensor but that the principle of strain equivalence permits a reduction to a tensor of order four the actual number of independent damage parameters in such a tensor is related to the material and damage symmetry for the isotropic case there must be two independent damage parameters which can be expressed in terms of damage parameter with physical meaning

on the computational significance of the strain space formulation of plasticity theory the strain space and traditional stress space formulations of plasticity theory are compared we show how plasticity theory has been implemented numerically to solve static and dynamic problems and conclude that the numerical loading criteria that have been used for two decades are identical to the strain space loading criteria proposed by naghdi and co workers

an adaptive wavelet galerkin method for an elastic plastic damage constitutive model 1d problem we present a wavelet galerkin method for the analysis of elastic plastic damage systems ju s model j w ju int j solids struct 25 1989 803 is explicitly considered and numerically tested but the method applies equally well to a broad range of damage mechanics models a three step elastic predictor plastic corrector damage corrector strategy is the core of the proposed approach we use a displacement approach with wavelet interpolating functions instead of classical polynomials this allows to clearly locate those sites in the system where damage is taking place thanks to the high pass filtering action operated by the wavelets this opens the way to inexpensive self adaptive strategies that are introduced and preliminary tested herein

on failure indicators in multidissipative materials abstract multi dissipative constitutive descriptions of irreversible material degradation result in tangent operators that are made up of multiple rank one updates of the elasticity tensors multisurface elastoplasticity plastic yielding combined with elastic degradation and multicrack models are representative examples the spectral properties of these tangent operators determine failure conditions at the material level in terms of loss of uniqueness and discontinuous bifurcation of the incremental response these failure properties are analysed herein by studying the eigensolution of the sum of n rank one m m matrices and the eigensolution of n rank one updates of the m m identity matrix whereby the tangent material tensors are written in matrix form analytical eigensolutions are presented and interpreted mechanically in terms of continuous and discontinuous failure indicators in the light of these spectral properties the failure indicators of single dissipative materials are revisited and explicit results are presented for double dissipative models in the form of plasticity combined with elastic damage in particular it is shown that the activation and interaction of two dissipation processes may destabilize the tangent operators beyond the state resulting from a single active mechanism

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

approximations to solitary waves on lattices ii quasi continuum methods for fast and slow waves a number of highly accurate analytic approximations to solitary waves on lattices are detailed and the results of numerical tests presented two types of approximation are considered the first set of approximations are generalizations of the continuum approximation based around various pade expansions of the operator these are convergent to the exact solution in the limit of wavespeed c to 1 or a dense lattice the author also considers ways of constructing approximations to fast solitary waves from a continuum type theory the two types are closely related in both cases significant improvements are made on existing approximations a symplectic hamiltonian integration scheme is used to perform computer simulations a new method of measuring the accuracy of a predicted waveform is also used on new and old approximations alike to compare the various methods considered

a general stochastic dynamic model of continuum damage mechanics abstract a stochastic dynamic probabilistic model of continuum damage mechanics is generalized from which various damage processes could be described in the same mathematical scheme the approach makes use of the deterministic damaged constitutive equations and introduces the randomized damage variable into the thermodynamic potential and the potential of dissipation in place of the deterministic ones an ito or stratonovich stochastic differential equation is derived based on the concept of an abstract dynamic system to describe the practical system under consideration a solution as a diffusion process can be obtained in the light of the theory of the stochastic differential equation the proposed model can be used to describe the state and stochastic evolution of various damage processes in relation to the irreversible rearrangements of micro structures of a solid from the proposed model both the stochastic characteristic of damage and the deterministic properties of damage embodied in the deterministic theory such as the non linearity of damage with regard to time or number of cycles und the effect of stress triaxiality on damage can be modelled

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