a geometrically nonlinear fe approach for the simulation of strong and weak discontinuities abstract in the present contribution a discontinuous finite element method for the computational modelling of strong and weak discontinuities in geometrically nonlinear elasticity is introduced the location of the interface is independent of the mesh structure and therefore discontinuous elements are introduced to capture the jump in the deformation map or its gradient respectively to model strong discontinuities the cohesive crack concept is adopted the inelastic material behaviour is covered by a cohesive constitutive law which associates the cohesive tractions acting on the crack surfaces with the jump in the deformation map in the case of weak discontinuities an extended nitsche s method is applied which ensures the continuity of the deformation map in a weak sense the applicability of the proposed method is highlighted by means of numerical examples dealing with both crack propagation and material interfaces

geometrically non linear and consistently linearized embedded strong discontinuity models for 3d problems with an application to the dissection analysis of soft biological tissues three different finite element formulations with embedded strong discontinuities are derived on the basis of the enhanced assumed strain method according to the work by jirasek and zimmermann int j numer methods engrg 50 2001 1269 they are referred to as statically optimal symmetric sos kinematically optimal symmetric kos and statically and kinematically optimal non symmetric skon formulations the effect of the discontinuities are characterized by additional degrees of freedom on the element level modifications to the standard kos and skon formulations are proposed in order to achieve consistency with the employed type of a three field hu washizu principle under mode i condition under this condition the formulation satisfies the internal compatibility at the discontinuity i e the relation between the stress in the bulk material and the traction across the discontinuity surface which is not the case for the classical kos formulation we propose a suitable explicit expression for a transversely isotropic traction law in form of a displacement energy function and assume that softening phenomena in the cohesive zone are modeled by a damage law which depends on the maximum gap displacement of the deformation path a linearization of all quantities which are related to the non linear problem leads to new closed form expressions in particular we focus attention on the linearization of the cohesive traction vector the associated element residua and stiffness matrices are provided standard static condensation of the internal degree of freedom leads to a generalized displacement model a comparative study of the modified formulations carried out by means of two numerical examples show the performance of the individual approach we employ constant strain tetrahedral elements with a single discontinuity embedded among the known stress locking phenomena associated with the sos formulation we recognized that the non symmetric skon formulation was not able to provide meaningful results for the dissection process of an arterial layer in three dimensions on distorted meshes for both numerical examples the symmetric kos formulation seems to be most suitable for representing the embedded discontinuities

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

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

integrable counterparts of the d kaup newell soliton hierarchy two integrable counterparts of the d kaup newell soliton hierarchy are constructed from a matrix spectral problem associated with the three dimensional special orthogonal lie algebra so 3 r an application of the trace identity presents hamiltonian or quasi hamiltonian structures of the resulting counterpart soliton hierarchies thereby showing their liouville integrability i e the existence of infinitely many commuting symmetries and conserved densities the involved hamiltonian and quasi hamiltonian properties are shown by computer algebra systems

aspects of the formulation and finite element implementation of large strain isotropic elasticity the paper presents a formulation of isotropic large strain elasticity and addresses some computational aspects of its finite element implementation on the theoretical side an eulerian setting of isotropic elasticity is discussed exclusively in terms of the finger tensor as a strain measure noval aspects are a direct representation of the eulerian elastic moduli in terms of the finger tensor and their rigorous decomposition into decoupled volumetric and isochoric contributions based on a multiplicative split of the finger tensor into spherical and unimodular parts the isochoric stress response is formulated in terms of the eigenvalues of the unimodular part of the finger tensor a constitutive algorithm for the computation of the stresses and tangent moduli for plane problems is developed and applied to a model problem of rubber elasticity on the computational side the implementation of the constitutive model in three possible finite element formulations is discussed after pointing out algorithmic techniques for the treatment of incompressible elasticity several numerical simulations are presented which show the performance of the proposed constitutive algorithm and the convergence behaviour of the different finite element fomulations for compressible and incompressible elasticity

on certain new nonlinear retarded integral inequalities in two independent variables and applications abstract the purpose of this paper is to establish some new non linear retarded integral inequalities in two independent variables which can be used as handy tools to study the boundedness of solutions of differential integral equations with the initial conditions an application is given to illustrate the usefulness of our results

a spectral problem based on so 3 r and its associated commuting soliton equations we present a spectral problem based on the three dimensional real special orthogonal lie algebra so 3 r and construct a hierarchy of commuting bi hamiltonian soliton equations by zero curvature equations associated with the spectral problem an illustrative example of soliton equations is computed together with its associated bi hamiltonian structure

the algebraic structures of isospectral lax operators and applications to integrable equations for a general spectral operator the author establishes types of algebraic structures of the spaces of the corresponding isospectral lax operators which essentially form the theoretical basis of the lax operator method furthermore the author introduces the concepts of tau algebras and master algebras to describe time polynomial dependent symmetries of nonlinear integrable equations finally the author applies the theory of lax operators to the kp hierarchy of integrable equations as an illustrative example and thus obtain the master symmetry algebra of the kp hierarchy

on the discrete constitutive models induced by strong discontinuity kinematics and continuum constitutive equations on the basis of continuum constitutive models stress vs strain the introduction of strong discontinuity kinematics considering jumps in the displacement fields across a discontinuity interface induces projected discrete constitutive models traction displacement jumps in a consistent manner therefore this projection provides possible links between the classical continuum strain localization analysis and the non linear decohesive fracture mechanics techniques the strong discontinuity analysis shows that bandwidth based regularization of the hardening softening parameter is the crucial modification to be done on the continuum model to achieve such a projection and it also provides the strong discontinuity conditions that set restrictions on the stress state compatible with bifurcations in a strong discontinuity format the methodology is illustrated on the basis of two classical families of non linear constitutive models scalar continuum damage and elasto plasticity for which the corresponding discrete constitutive models and the strong discontinuity conditions are derived

on some new inequalities related to certain inequalities in the theory of differential equations in the present paper we establish some new integral inequalities related to a certain inequality arising in the theory of differential equations the discrete analogues of the main results are also given the analysis used in the proofs is elementary and the results established here provide new estimates for these types of inequalities

a soliton hierarchy associated with a new spectral problem and its hamiltonian structure a hierarchy of soliton equations together with its hamiltonian structure is constructed from a new spectral problem associated with the three dimensional special orthogonal real lie algebra so 3 the liouville integrability of the presented soliton hierarchy is proved based on the hamiltonian structure

generalization of a retarded gronwall like inequality and its applications abstract this paper generalizes a lipovan s result of gronwall like inequalities j math anal appl 252 2000 389 to a new type of retarded inequalities which includes both a nonconstant term outside the integrals and more than one distinct nonlinear integrals from our result bihari s result and pinto s result can be deduced as some special cases our result can be applied to give the global existence and estimate solutions we also apply it to improve an estimation for almost periodicity of an invariant manifold

on generalized of certain retarded nonlinear integral inequalities and its applications in retarded integro differential equations in this paper some new nonlinear retarded integral inequalities of gronwall bellman pachpatte type are investigated some applications are also presented to illustrate the usefulness of some of our results in estimation of solution of certain retarded integro differential equations with the initial conditions

two integrable couplings of a generalized d kaup newell hierarchy and their hamiltonian and bi hamiltonian structures abstract we enlarge the spectral problem of a generalized d kaup newell d kn spectral problem solving the enlarged zero curvature equations we produce integrable couplings a reduction of the spectral matrix leads to a second integrable coupling system next bilinear forms that are symmetric ad invariant and non degenerate on the given non semisimple matrix lie algebra are computed to employ the variational identity the variational identity is then applied to the original enlarged spectral problem of a generalized d kn hierarchy and the reduced problem hamiltonian structures are presented as well as a bi hamiltonian formulation of the reduced problem both hierarchies have infinitely many commuting high order symmetries and conserved densities i e are liouville integrable

lax representations and lax operator algebras of isospectral and nonisospectral hierarchies of evolution equations a method of constructing lax representations for isospectral and nonisospectral hierarchies of evolution equations is proposed it is shown that under some simple but essential conditions the corresponding lax operators may constitute infinite dimensional lie algebras with respect to the binary operation given in this paper furthermore the detailed analyses for the kdv couple the akns couple and a new couple of isospectral and nonisospectral hierarchies of integrable equations are presented as examples

some nonlinear gronwall bellman type retarded integral inequalities with power and their applications abstract the purpose of this paper is to establish some new retarded integral inequalities of gronwall bellman type which generalizes some known integral inequalities we use two lemmas to simplify the complex inequalities and give an upper bound estimation of the unknown function the inequalities given here can be used in the analysis of the qualitative properties of certain classes of differential equations and integral equations we apply our results to the boundedness of the solutions of integral equations and differential system

3d crack propagation in unreinforced concrete a two step algorithm for tracking 3d crack paths tensile failure of unreinforced concrete involves progressive micro cracking and the related strain softening can coalesce into geometrical discontinuities which separate the material advanced mechanical theories and numerical schemes are required to efficiently and adequately represent crack propagation in 3d in this paper we use the concept of strong discontinuities to model concrete failure we introduce a cohesive fracture process zone which is characterized by a transversely isotropic traction separation law we combine the cohesive crack concept with the partition of unity finite element method where the finite element space is enhanced by the heaviside function the concept is implemented for tetrahedral elements and the failure initialization is based on the simple non local rankine criterion for each element we assume the embedded discontinuity to be flat in the reference configuration which leads to a non smooth crack surfaces approximation in 3d in general different concepts for tracking non planar cracks in 3d are reviewed in addition we propose a two step algorithm for tracking the crack path where a predictor step defines discontinuities according to the non local failure criterion and a corrector step draws in non local information of the existing discontinuities in order to predict a closed 3d crack surface implementation details are provided the proposed framework is used to analyze the predictability of concrete failure by two benchmark examples i e the nooru moharned test and the brokenshire test we compare our numerical results which are mesh independent with experimental data and numerical results adopted from the literature

a soliton hierarchy associated with so 3 r we generate a hierarchy of soliton equations from zero curvature equations associated with the real lie algebra so 3 r and show that each equation in the resulting hierarchy has a bi hamiltonian structure and thus integrable in the liouville sense

on the localization properties of multiplicative hyperelasto plastic continua with strong discontinuities abstract the objective of this work is to examine the large strain localization properties of hyperelasto plastic materials which are based on the multiplicative decomposition of the deformation gradient thereby the case of strong discontinuities is investigated to this end first an explicit expression for the spatial tangent operator is given taking into account anisotropic as well as nonassociated material behaviour then the structure of a regularized discontinuous velocity gradient is elaborated and discussed in detail based on these two results the localization condition is derived with special emphasis on the loading conditions inside and outside an anticipated localization band thereby the intriguingly simple structure of the tangent operator which resembles the structure of the geometrically linear theory is extensively exploited this similarity carries over to the general representation for the critical hardening modulus which is exemplified for isotropic materials as a result analytical solutions are available under the assumption of small elastic strains which is justified for metals finally examples are given for the special case of the associated von mises flow rule to this end the critical localization direction and the critical hardening modulus are investigated with respect to the amount of finite elastic strain within different modes of homogeneous elasto plastic deformations

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

finite elements with displacement interpolated embedded localization lines insensitive to mesh size and distortions a new finite element formulation aimed at the solution of problems involving strain localization is presented the proposed formulation incorporates displacement interpolated embedded localization lines results are shown to converge to an exact solution when the mesh is refined and also to be quite insensitive to mesh distortions

a framework for fracture modelling based on the material forces concept with xfem kinematics a theoretical and computational framework which covers both linear and non linear fracture behaviour is presented as a basis for the formulation we use the material forces concept due to the close relation between on one hand the eshelby energy momentum tensor and on the other hand material defects like cracks and material inhomogeneities by separating the discontinuous displacement from the continuous counterpart in line with the extended finite element method xfem we are able to formulate the weak equilibrium in two coupled problems representing the total deformation however in contrast to standard xfem where the direct motion discontinuity is used to model the crack we rather formulate an inverse motion discontinuity to model crack development the resulting formulation thus couples the continuous direct motion to the inverse discontinuous motion which may be used to simulate linear as well as non linear fracture in one and the same formulation in fact the linear fracture formulation can be retrieved from the non linear cohesive zone formulation simply by confining the cohesive zone to the crack tip these features are clarified in the two numerical examples which conclude the paper copyright 2005 john wiley sons ltd

improved versions of assumed enhanced strain tri linear elements for 3d finite deformation problems abstract improved three dimensional tri linear elements for finite deformation problems are developed based on an assumed enhanced strain methodology which in the linear regime incorporates the classical method of incompatible modes as a particular case three crucial modifications of a recently proposed element which reduces to wilson s brick in the linear regime are introduced to prevent locking response in distorted configurations and to maintain proper rank while preserving excellent performance in bending dominated and localization problems i a modified quadrature rule ii an additional enhancement of the divergence term and iii a modification of the isoparametric shape function derivatives for the three dimensional problem moreover these modified shape function derivatives are shown to improve the performance of the standard tri linear brick in distorted configurations in addition a strategy is described to circumvent the memory storage requirements in the static condensation procedure of the enhanced strain parameters the excellent performance of the improved methodology is illustrated in representative numerical simulations

a retarded gronwall like inequality and its applications we prove a gronwall like inequality and present some of its applications to the qualitative study of retarded differential equations the problems of global continuation of the solutions and the existence of nonoscillatory solutions are considered by means of examples we show the usefulness of our results

bihamiltonian geometry darboux coverings and linearization of the kp hierarchy we use ideas of the geometry of bihamiltonian manifolds developed by gel fand and zakharevich to study the kp equations in this approach they have the form of local conservation laws and can be traded for a system of ordinary differential equations of riccati type which we call the central system we show that the latter can be linearized by means of a darboux covering and we use this procedure as an alternative technique to construct rational solutions of the kp equations

an integrable generalization of the d kaup newell soliton hierarchy and its bi hamiltonian reduced hierarchy abstract we present a new spectral problem a generalization of the d kaup newell spectral problem associated with the lie algebra sl 2 r zero curvature equations furnish the soliton hierarchy the trace identity produces the hamiltonian structure for the hierarchy and shows its liouville integrability lastly a reduction of the spectral problem is shown to have a different soliton hierarchy with a bi hamiltonian structure the major motivation of this paper is to present spectral problems that generate two soliton hierarchies with infinitely many conservation laws and high order symmetries

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

some new nonlinear inequalities and applications to boundary value problems abstract in this paper we establish some new nonlinear integral inequalities of the gronwall bellman ou iang type in two variables these on the one hand generalizes and on the other hand furnish a handy tool for the study of qualitative as well as quantitative properties of solutions of differential equations we illustrate this by applying our new results to certain boundary value problem

estimation of unknown function of a class of integral inequalities and applications in fractional integral equations in this paper we establish a class of iterated integral inequalities which includes a nonconstant term outside the integrals the upper bound of the embedded unknown function is estimated explicitly by adopting novel analysis techniques such as change of variable amplification method differential and integration the derived result can be applied in the study of qualitative properties of solutions of fractional integral equations

modeling strong discontinuities at finite strains a novel numerical implementation abstract in this paper a finite element formulation suitable for the modeling of locally embedded strong discontinuities at finite strains is presented following the enhanced assumed strain concept eas the proposed numerical model is based on an additive decomposition of the displacement gradient into a conforming and an enhanced part the enhanced part is associated with the final failure kinematics of solids which are approximated by means of a discontinuous displacement field strong discontinuities referring to the displacement jump no special assumption such as purely mode i or mode ii failure is made the same holds for the class of interface laws considered which govern the evolution of the displacement discontinuity in terms of the traction vector acting at the surface of strong discontinuities consequently the suggested numerical framework can be applied to a broad range of different interface laws including damage based models in contrast to previous works the presented finite element formulation does not require the static condensation technique to be employed more precisely instead of computing the conforming part of deformation and the displacement jump simultaneously from the weak form of equilibrium and the weak form of traction continuity the different parts of the local deformation are decomposed according to a predictor corrector algorithm the proposed predictor and the corrector step are formally identical to that of classical computational plasticity models hence subroutines designed for standard models continuous deformation can be applied with only minor modifications necessary the applicability as well as the performance of the resulting finite element formulation are demonstrated by means of a fully three dimensional analysis of shear band formation in a bar made of a ductile material

on some new integral inequalities of gronwall bellman pachpatte type abstract in this paper we establish new nonlinear integral inequalities of gronwall bellman pachpatte type these inequalities generalize some former famous inequalities and can be used as handy tools to study the qualitative as well as the quantitative properties of solutions of some nonlinear ordinary differential and integral equations the purpose of this paper is to extend certain results which proved by pachpatte in inequalities for differential and integral equations academic press new york and london 1998 some applications are also given to illustrate the usefulness of our results

modelling strong discontinuities in solid mechanics via strain softening constitutive equations part 2 numerical simulation on the basis of the strong discontinuity analysis of standard local stress strain constitutive equations a finite element framework for the simulation of strong discontinuities which belongs to the family of assumed enhanced strain methods is presented taking the standard linear triangle as the underlying element an additional incompatible mode leads to the formulation of an enriched strain field which is shown to be able to appropriately capture strong discontinuities the presented numerical simulations show that mesh size and mesh alignment dependencies can be completely removed

on liouville integrability of zero curvature equations and the yang hierarchy sufficient conditions for a zero curvature equation ut vx u v 0 being liouville integrable are investigated in the case that the equation is integrable an explicit formula of the poisson bracket h lambda h mu for hamiltonians h is proposed the yang hierarchy is derived and shown to be liouville integrable

strong discontinuity approaches an algorithm for robust performance and comparative assessment of accuracy abstract the strong discontinuity approach sda is a popular method to incorporate cracks as displacement discontinuities into finite elements in the first part of the paper different sda formulations denoted as sos kos and skon are assessed numerically based upon different error norms including a norm to evaluate the variational consistency of sda elements results are compared with analytical solutions as well as with results from interface elements and the element erosion technique for cohesionless cracks in the second part of the paper a new method to improve the computational robustness of sda analyses is proposed in addition to using arc length control to solve the nonlinear equation system a sequential un and reloading scheme is proposed with the crack state frozen during unloading to avoid that more than one new crack segment is activated within an increment the performance of the algorithm is demonstrated by numerical analyses of a tension test on a dog bone shaped specimen by means of different variants of the sda and for comparison also using interface elements

a comparative study on finite elements for capturing strong discontinuities e fem vs x fem a comparative study on finite elements for capturing strong discontinuities by means of elemental e fem or nodal enrichments x fem is presented based on the same constitutive model continuum damage and linear elements triangles and tetrahedra optimized implementations of both types of enrichments in the same non linear code are tested for a representative set of 2d and 3d crack propagation examples it is shown that both methods provide the same qualitative and quantitative results for enough refined meshes for the performed tests e fem exhibited in general a higher accuracy mostly for coarse meshes whereas convergence rate with mesh refinement which is super linear showed slightly higher for x fem as for the computational costs for single crack modelling x fem showed depending on the case from 1 1 to about 2 5 times more expensive than e fem for multiple cracks the computational cost of e fem keeps constant whereas the cost associated to x fem increases linearly with the number of modelled cracks

a coupled akns kaup newell soliton hierarchy a coupled akns kaup newell hierarchy of systems of soliton equations is proposed in terms of hereditary symmetry operators resulted from hamiltonian pairs zero curvature representations and tri hamiltonian structures are established for all coupled akns kaup newell systems in the hierarchy therefore all systems have infinitely many commuting symmetries and conservation laws two reductions of the systems lead to the akns hierarchy and the kaup newell hierarchy and thus those two soliton hierarchies also possess tri hamiltonian structures

robust and direct evaluation of j2 in linear elastic fracture mechanics with the x fem numerical simulation of fracture mechanics is of major importance in the design of structures such as planes or dams for such complex structures robustness and accuracy of the simulations are necessary the finite element method is classically considered for the simulation of such problems however the crack must be explicitly meshed in order to incorporate the discontinuity of the displacement field across crack faces this process has to be handled at each step of propagation the main drawback of this approach is the computational and human cost of remeshing the x fem method was proposed as a response to this issue 1 2 it uses the concept of partition of unity 3 in two ways first to take into account the displacement jump across crack faces far away from the crack tip and also to enrich the approximation close to the tip with some analytical asymptotic fields another important issue in fracture mechanics is the prediction of crack growth direction several criteria have been proposed one can cite for example the maximum hoop stress criterion 4 the maximum tangential stress criterion or the symmetry principle 5 the maximum energy release rate criterion 6 also called vectorial j integral criterion 7 is an alternative which was initially proposed by hellen and blackburn 7 the main difference in evaluating this criterion with respect to the classical j integral rely on the difficulties in the evaluation of the second component j2 of the energy release rate vector j it is well known that j1 which is basically the j integral is path independent concerning j2 herrmann and herrmann 8 have shown that it is path independent only in a modified manner i e by adding an extra contribution on the crack faces however this new contribution requires an accurate determination of the mechanical fields near the crack tip it is well known that the quality of these fields near a singularity is not ensured in the finite element context this is why some authors proposed specific strategies to avoid the integration near the tip for an accurate numerical evaluation of j2 9 10 for enriched finite