regularization of microplane damage models using an implicit gradient enhancement abstract the microplane model allows for the description of damage induced anisotropy in a natural manner by introducing constitutive laws for quantities on individual microplanes at each material point however if damage or other strain softening constitutive laws are used within the microplane approach the well known problem of localization arises leading to spurious results and mesh dependency this problem demands some regularization method to stabilize the solution the paper focuses on the efficient implementation of implicit gradient enhancement for microplane damage models previous works enhanced the strain tensor thus resulting in large number of extra degrees of freedom which limits the use of this method for large scale 3d simulations a new method which enhances the equivalent strain driving the damage on each microplane is introduced in this work the new method limits the number of additional degrees of freedom to one while preserving the regularizing effect the two methods are implemented within a 3d finite element code to compare their performance the microplane model used is based on a thermodynamically consistent formulation and on a volumetric deviatoric split of strains on each microplane furthermore an exponential damage law is used and an equivalent strain expression which distinguishes between compression and tension is applied to simulate the behavior of concrete the capabilities of the proposed formulation are demonstrated by comparison to published experiments on plain concrete

over nonlocal gradient enhanced plastic damage model for concrete abstract classical continuum models exhibit strong mesh dependency during softening one method to regularize the problem is to introduce a length scale parameter via the nonlocal formulation however standard nonlocal enhancement either by integral or gradient formulation may serve only as a partial localization limiter for many material models the over nonlocal formulation where the weight for the nonlocal value is greater than unity and the excesses compensated by assigning a negative weight to the local value is able to fully regularize certain material models when standard nonlocal enhancement fails to do so a plastic damage model for concrete is formulated with this over nonlocal enhancement via the gradient approach and the full regularizing capabilities demonstrated

damage integration in the strain space the focus is on isotropic elastodamaging softening materials where the damage parameter is expressed as a function of the total strain by integrating the mechanical work in the strain space along a stepwise holonomic loading history an incremental strain energy is obtained a coaxiality condition for the incremental strain energy to be potential is identified and its implications on the associativity of the damage evolution are discussed under some hypotheses the increment of the mechanical work is shown to be minimum along strain radial paths these results are used to construct a multifield variational framework supporting finite element nonlocal formulations

a damage model with evolving nonlocal interactions we present a damage model for softening materials with evolving nonlocal interactions the thermodynamic implications and the material stability issue are addressed the proposed nonlocal averaging scheme provides the obtained constitutive models with an evolving nonlocal interaction which is activated only when damage occurs in the analysis of structures made of quasi brittle materials this feature helps not only to overcome some issues with the incorrect initiation of damage but also to better control the evolving size of the active fracture process zone this is an essential feature that is usually not considered in depth in many existing nonlocal approaches to the continuum modelling of quasi brittle fracture numerical examples are given to demonstrate features of the proposed modelling approach

adaptive modeling of damage growth using a coupled fem bem approach summary we propose a coupled boundary element method bem and a finite element method fem for modelling localized damage growth in structures bem offers the flexibility of modelling large domains efficiently while the non linear damage growth is accurately accounted by a local fem mesh an integral type nonlocal continuum damage mechanics with adapting fem mesh is used to model multiple damage zones and follow their propagation in the structure strong form coupling bem hosted is achieved using lagrange multipliers because the non linearity is isolated in the fem part of the system of equations the system size is reduced using schur complement approach then the solution is obtained by a monolithic newton method that is used to solve both domains simultaneously the coupled bem fem approach is verified by a set of convergence studies where the reference solution is obtained by a fine fem in addition the method is applied to multiple fractures growth benchmark problems and shows good agreement with the literature copyright 2015 john wiley sons ltd

the discontinuity enriched finite element method we introduce a new methodology for modeling problems with both weak and strong discontinuities independently of the finite element discretization at variance with the extended generalized finite element method x gfem the new method named the discontinuity enriched finite element method de fem adds enriched degrees of freedom only to nodes created at the intersection between a discontinuity and edges of elements in the mesh although general the method is demonstrated in the context of fracture mechanics and its versatility is illustrated with a set of traction free and cohesive crack examples we show that de fem recovers the same rate of convergence as the standard fem with matching meshes and we also compare the new approach to x gfem

computational modeling of localized failure in solids xfem vs pf czm abstract this work addressesnumerical comparison between the extended finite element method xfem and the phase field regularized cohesive zone model pf czm wu 2017 2018a for the modeling of cohesive fracture induced localized failure in solids a novel implementation of the pf czm is proposed similarly to the xfem in which the added degrees of freedom characterizing the crack behavior are associated only with those nodes within a small sub domain representative benchmark examples show that both methods are independent of the mesh discretization for problems with the crack path known a priori both methods give almost coincident numerical results though the xfem is advantageous due to its high coarse mesh resolution for complex problems with arbitrary crack propagation even though some discrepancies are observed the numerical results given by both methods are also quantitatively similar if the predicted crack trajectories are close to each other despite the higher computational cost in this case the pf czm is much more favored due to its intrinsic capability of modeling complex crack configurations e g nucleation propagation merging and branching etc with no need of any extra strategy the above facts validate our previous theoretical analysis that the pf czm approaches asymptotically to a discontinuous czm for a vanishing length scale as it avoids most challenging issues exhibited by other approaches the pf czm is not only theoretically elegant but also numerically appealing for the modeling of localized failure in solids

a consistent point searching algorithm for solution interpolation in unstructured meshes consisting of 4 node bilinear quadrilateral elements to translate and transfer solution data between two totally different meshes i e mesh 1 and mesh 2 a consistent point searching algorithm for solution interpolation in unstructured meshes consisting of 4 node bilinear quadrilateral elements is presented in this paper the proposed algorithm has the following significant advantages 1 the use of a point searching strategy allows a point in one mesh to be accurately related to an element containing this point in another mesh thus to translate transfer the solution of any particular point from mesh 2 td mesh 1 only one element in mesh 2 needs to be inversely mapped this certainly minimizes the number of elements to which the inverse mapping is applied in this regard the present algorithm is very effective and efficient 2 analytical solutions to the local co ordinates of any point in a four node quadrilateral element which are derived in a rigorous mathematical manner in the context of this paper make it possible to carry out an inverse mapping process very effectively and efficiently 3 the use of consistent interpolation enables the interpolated solution to be compatible with an original solution and therefore guarantees the interpolated solution of extremely high accuracy after the mathematical formulations of the algorithm are presented the algorithm is tested and validated through a challenging problem the related results from the test problem have demonstrated the generality accuracy effectiveness efficiency and robustness of the proposed consistent point searching algorithm copyright c 1999 john wiley sons ltd

high performance parallel simulation of structure degradation using non local damage models simulating damage and failure of metallic or composite structures often fails when using standard finite element discretizations and a newton raphson solving procedure the difficulties arise from an uncontrolled mesh dependency caused by damage localization to structural instabilities and to an increase of computational costs the first problem can be overcome by using non local damage constitutive equations together with a specific finite element model the second problem can then be solved with an arc length method which manages instabilities and snap backs they are caused by the loss of strength carrying capacity of the damaged material finally the whole computational scheme is parallelized the main contribution of this paper consists in bringing together the three numerical techniques non local material model piloting and parallel computations the resulting numerical scheme allows to go beyond the overly simplistic cases often encountered in the literature it allows robust industrial simulations with a high number of degrees of freedom both in two and three dimensions and deepened studies involving a large number of simulations

seshadri constants and periods of polarized abelian varieties the purpose of this paper is to study the seshadri constants of abelian varieties consider a polarized abelian variety a l of dimension g over the field of complex numbers one can associate to a l a real number e a l its seshadri constant which in effect measures how much of the positivity of l can be concentrated at any given point of a the number e a l can be defined as the rate of growth in k of the number of jets that one can specify in the linear series oa kl alternatively one considers the blow up f x blx x x of x at a point x with exceptional divisor e x over x and defines e a l def sup e r f l ee is nef

remark on the decay for damped string and beam equations where a is an operator in a hilbert space h m is a real function our results improve those obtained by de brito in a recent paper l we briefly recall the notation and the assumptions in l referring to this paper for other information on physical meaning existence and regularity theory applications and bibliography let a be a self adjoint positive operator in a real hilbert space h with dense domain v we assume that 5 0 is the least eigenvalue of a so a u au u 3 51 1 for u e v let m e cl be a real nondecreasing function such that m s 2 p ks k z 0 p real we have with the notation

gradient enhanced damage for quasi brittle materials summary conventional continuum damage descriptions of material degeneration suffer from loss of well posedness beyond a certain level of accumulated damage as a consequence numerical solutions are obtained which are unacceptable from a physical point of view the introduction of higher order deformation gradients in the constitutive model is demonstrated to be an adequate remedy to this deficiency of standard damage models a consistent numerical solution procedure of the governing partial differential equations is presented which is shown to be capable of properly simulating localization phenomena

strain based transient gradient damage model for failure analyses a transient gradient enhanced damage model has been developed for the numerical modelling of the damage and fracture process within a continuum damage mechanics framework some deficiencies of existing gradient enhanced damage formulations for the simulation of macroscopic crack propagation are pointed out the transient gradient approach assumes a direct coupling between the material length parameter and the local strain state of the material which leads to a transient behaviour of the nonlocal effect details of the method are presented and fully elaborated in an incremental iterative solution scheme mesh objectivity and physical relevance of the method are analysed by one dimensional and two dimensional numerical examples

vector bundles of rank 2 and linear systems on algebraic surfaces let s be a smooth complex algebraic surface and let l be a divisor on s it is well known that the linear system il ksi where ks is the canonical divisor plays an important role in understanding the geometry of s several authors see for example 8 11 studied various properties of this system our paper treats the same subject but the point of view is different from the works mentioned above the philosophy underlying our approach is almost classical points on s more generally effective 0 cycles in special position with respect to il ksi see definition below contain information about the geometry of s this point of view was recently revived in 5 also 10 in fact this paper contains the technique of going from the effective 0 cycle z in special position with respect to il ksi to the geometry of s for the sake of completeness we give a definition and state a theorem which can be found in 5 10

iterative lcp solvers for non local loading unloading conditions this paper deals with the finite element analysis of a certain class of non local dissipative constitutive models where the canonical pointwise backward euler scheme cannot be employed for satisfying the loading unloading conditions in the presence of a non local dissipation the admissibility conditions in a point depend on the inelastic strain increment of the surrounding points and can be cast as a linear complementarity problem lcp involving all gauss points of the process zone in order to actually solve the lcp the use of iterative algorithms that can be easily embodied into existing fe codes is discussed the performance of the proposed algorithms is tested in 1d and 2d examples for both elastoplastic and damaging materials copyright 2003 john wiley sons ltd

seshadri constants on abelian varieties we show that on a complex abelian variety of dimension two or greater the seshadri constant of an ample line bundle is at least one moreover the seshadri constant is equal to one if and only if the polarized abelian variety splits as a product of a principally polarized elliptic curve and a polarized abelian subvariety of codimension one we also examine the case when the seshadri constant is not one and obtain lower bounds when the dimension of the abelian variety is small

multiscale computational homogenization methods with a gradient enhanced scheme based on the discontinuous galerkin formulation abstract when considering problems of dimensions close to the characteristic length of the material the size effects can not be neglected and the classical so called first order multiscale computational homogenization scheme fmch looses accuracy motivating the use of a second order multiscale computational homogenization smch scheme this second order scheme uses the classical continuum at the micro scale while considering a second order continuum at the macro scale although the theoretical background of the second order continuum is increasing the implementation into a finite element code is not straightforward because of the lack of high order continuity of the shape functions in this work we propose a smch scheme relying on the discontinuous galerkin dg method at the macro scale which simplifies the implementation of the method indeed the dg method is a generalization of weak formulations allowing for inter element discontinuities either at the c 0 level or at the c 1 level and it can thus be used to constrain weakly the c 1 continuity at the macro scale the c 0 continuity can be either weakly constrained by using the dg method or strongly constrained by using usual c 0 displacement based finite elements therefore two formulations can be used at the macro scale i the full discontinuous galerkin formulation fdg with weak c 0 and c 1 continuity enforcements and ii the enriched discontinuous galerkin formulation edg with high order term enrichment into the conventional c 0 finite element framework the micro problem is formulated in terms of standard equilibrium and periodic boundary conditions a parallel implementation in three dimensions for non linear finite deformation problems is developed showing that the proposed method can be integrated into conventional finite element codes in a straightforward and efficient way

damage growth modeling using the thick level set tls approach efficient discretization for quasi static loadings the thick level set tls model is a damage model containing a non local treatment that prevents from spurious localization issues it also offers an automatic transition from damage to fracture the tls approach to model damage growth was first presented in moes et al 24 with a first numerical implementation for a time dependent damage evolution law in this paper we propose some improvements in terms of discretization and explicit damage growth algorithms to obtain a robust efficient and easy to implement model these improvements include a simple and efficient variational formulation for computing the non local quantities as well as the introduction of a so called ramped heaviside enrichment function to properly take into account the transition to cracks in fully damaged zones we consider here a simple explicit formulation for quasi static loadings decoupling the elastic computation and the damage growth the method is validated through standard benchmarks and compared to the griffith s fracture theory the convergence of energy and displacement errors is observed more complex computations including damage initiations are eventually performed

a new marching ridges algorithm for crack path tracking in regularized media abstract tracking algorithms are used to predict crack paths in structures modeled with the finite element method in such a way that the paths do not depend on the selected mesh for regularized media the simplest methods rely on scalar variables somehow related to material degradation despite their simplicity they suffer from a major limitation they allow the crack to initiate and propagate in only one direction consequently such approaches usually fail in case of crack branching or crack initiation inside the structure to overcome this difficulty we propose a new crack path tracking algorithm it is designed to simultaneously detect several local maxima of a degradation related variable by following the associated ridge lines that is why the algorithm proposed in this paper could be referred to as a marching ridges algorithm the performance of the proposed approach is illustrated by three numerical examples within different frameworks the first ones show that the algorithm can be used to insert crack increments during a ductile failure computation with a quasi static implicit resolution procedure in 2d and 3d the last example proves that the algorithm can be used as a post processing tool to capture dynamic crack branching from a damage distribution image only

an improved stable xfem is xfem with a novel enrichment function for the computational modeling of cohesive cracks abstract owing to the solution space enriched with a priori known knowledge through the partition of unity the extended or generalized finite element method xfem gfem has been widely applied to the modeling of localized failure in solids as far as cohesive cracks are concerned in this work on the one hand the xfem enriched with heaviside function or the shifted one with respect to nodal values is of optimal accuracy but is prone to the issue of ill conditioned or even singular system matrix for arbitrary crack propagation on the other hand the stable xfem s xfem circumvents the ill conditioning issues by introducing the residual heaviside function with respect to its linear interpolant at the cost of losing accuracy e g spurious stress locking for discontinuities with non uniform displacement jumps aiming to reconcile the incompatibility between accuracy of the solution and conditioning of the system matrix this work addresses a simple yet effective improved stable xfem is xfem with a novel enrichment function both the xfem and s xfem are recovered as its particular cases with different approximation schemes for the bridging scale which incorporates information from both the continuous coarse scale and the discontinuous fine scale in the proposed is xfem as the enrichment basis functions are not contained in the standard polynomial space conditioning of the resulting system matrix is nearly insensitive to the mesh discontinuity configuration even if the crack path gets close to element nodes furthermore the is xfem with a small stabilization parameter is almost of the same accuracy as the xfem particularly pathological numerical results polluted by spurious stress locking exhibited in the s xfem are not observed across a fully softened discontinuity with non uniform displacement jumps as only the enrichment function is modified the proposed is xfem together with appropriate crack propagation criterion and tracking algorithm can be implemented easily in the existing xfem codes several representative numerical simulations of element and structure benchmark tests are presented to validate the proposed is xfem regarding both accuracy of the solution and conditioning of the system matrix

localizing gradient damage model with decreasing interactions summary nonlocal integral and or gradient enhancements are widely used to resolve the mesh dependency issue with standard continuum damage models however it is reported that whereas the structural response is mesh independent a spurious damage growth is observed accordingly a class of modified nonlocal enhancements is developed in literature where the interaction domain increases with damage in this contribution we adopt a contrary view that the interaction domain decreases with damage this is motivated by the fact that the fracture of quasi brittle materials typically starts as a diffuse network of microcracks before localizing into a macroscopic crack to ensure thermodynamics consistency the micromorphic theory is adopted in the model development the ensuing microforce balance resembles closely the helmholtz expression in a conventional gradient damage model the superior performance of the localizing gradient damage model is demonstrated through a one dimensional problem as well as mode i and ii failure in plane deformation for all three cases a localized deformation band at material failure is obtained this article is protected by copyright all rights reserved

phase field based nonlocal anisotropic damage mechanics model abstract a nonlocal anisotropic damage theory is developed in this work through the phase field method to address the anisotropic damage evolution in materials the anisotropic damage is discussed and appropriate nonconserved order parameters in three mutually perpendicular directions are defined to find the growth of the components of a second order diagonal damage tensor corresponding to the principal directions of a general second order damage tensor in contrast to the previous models two new tensors are proposed to act as interpolation and potential functions along with the allen cahn equation in order to obtain the evolution of the order parameters which is the basis of the definition of the damage rate the tensor formulation of the growth of the components of the damage tensor is proposed for the first time it is shown that by introducing a set of material parameters including a length scale parameter due to damage there is a robust and simplified way to model the nonlocal behavior of damage and predict the corresponding material behavior as components of a second order diagonal damage tensor

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

an isogeometric analysis approach to gradient damage models continuum damage formulations are commonly used for the simulation of diffuse fracture processes implicit gradient damage models are employed to avoid the spurious mesh dependencies associated with local continuum damage models the c 0 continuity of traditional finite elements has hindered the study of higher order gradient damage approximations in this contribution we use isogeometric finite elements which allow for the construction of higher order continuous basis functions on complex domains we study the suitability of isogeometric finite elements for the discretization of higher order gradient damage approximations

a thermodynamically motivated implicit gradient damage framework and its application to brick masonry cracking a new gradient damage model is derived starting from thermodynamic statements nonstandard terms are added to the free energy potential for this purpose in contrast with existing approaches these terms are related to the kinematics of the problem and therefore come into play already in the elastic regime following the argument that the second law of thermodynamics need only be satisfied globally for a nonlocal continuum an energy balance equation is derived which must be solved together with the usual equilibrium equations numerical solutions of this coupled set of equations have been generated for a one dimensional problem in order to study the localisation behaviour of the new model in the limit where the elastic modulus related to the energy balance goes to 0 an earlier gradient damage model is retrieved this simplified model is used to study cracking of brick masonry under the influence of mechanical loading

existence uniqueness and asymptotic behavior for solutions of the nonlinear beam equation a mathematical model for the transverse deflection of an extensible beam of length l whose ends are held at fixed distance apart is equation 2 u t 2 4 u x 4 0 l u 2 t d 2 u x 2 0 which has been proposed by woinowsky and krieger 18 where is a positive constant is a constant not necessarily positive and the nonlinear term represents the change in the tension of the beam due to its extensibility

a continuous discontinuous model for crack branching this is the peer reviewed version of the following article tamayo e et al a continuous discontinuous model for crack branching international journal for numerical methods in engineering 5 octubre 2019 vol 120 num 1 p 86 104 which has been published in final form at https doi org 10 1002 nme 6125 this article may be used for non commercial purposes in accordance with wiley terms and conditions for self archiving

a combined xfem damage mechanics approach for concrete crack propagation abstract this paper presents a crack model that couples the benefits of the damage mechanics approach and the extended finite element method xfem a crack tracking technique is developed to propagate the crack path along a single row of finite elements as a function of the stress based nonlocal method the level sets are computed to predict the crack path and the path is corrected with the use of continuum damage mechanics once a certain level of damage is reached inside an element the previously computed level sets are used to apply the xfem formulation a cohesive xfem formulation is used to transfer energy dissipation from the damage mechanics approach to the xfem model using numerous test cases it is shown that the model is mesh independent and that it removes the spurious stress transfer exhibited by continuum damage models various transition damage criteria are compared and the optimal one is determined

damage modeling employing strain gradient continuum theory a damage model for quasi brittle materials embedded into the two dimensional c1 continuity triangular finite element formulation based on the strain gradient continuum theory is considered the isotropic damage law is applied to the higher order stress strain constitutive model which enables the analysis of both homogeneous and heterogeneous materials such softening formulation also ensures a decrease of the intensity of the material nonlocality associated with the damage growth which is necessary for the correct description of the narrow localized deformation in order to obtain the required constitutive matrices the second order homogenization procedure is applied to the various representative volume elements in the frame of a multiscale approach the derived finite element formulation is implemented into the finite element program abaqus by means of user subroutines the superior regularization capabilities as well as the accuracy and efficiency of the proposed higher order gradient damage model are demonstrated by the standard benchmark examples

two gradient plasticity theories discretized with the element free galerkin method abstract in this paper the element free galerkin method is exploited to analyze gradient plasticity theories the use of this discretization method has the advantage that higher order continuity which can be required for the plastic multiplier field is provided for in passing we show that the regularization properties of the higher order gradients are necessary since similar to finite element methods a severe discretization sensitivity is encountered otherwise the mesh free discretization method is first applied to an established stress space gradient plasticity theory next a strain space gradient plasticity theory which employs an implicit averaging of an invariant strain measure is proposed and elaborated this theory is insensitive to the order of the interpolation polynomials and as a result the convergence of the newton raphson procedure is better the performance of both gradient plasticity models is examined for the mesh free discretization scheme a comparative study is carried out for a one dimensional bar in tension while the influence of the discretization in biaxial compression is studied next including that of the numerical parameters of the discretization scheme and that of the internal length scale contained in the gradient plasticity theories

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

gradient enhanced damage modeling of cracked bodies by reproducing kernel element method abstract in crack modeling by extended finite element method the enrichment by the asymptotic analytical solution works excellent for linear elastic fracture mechanics however for the more sophisticated condition of a gradient enhanced damage model firstly the analytical solution is not available and secondly higher order continuity of the shape functions is required in this study the extended finite element model of the cracked body is augmented with shape functions of higher order continuity and higher order consistency generated by reproducing kernel element method they are used in a region near the crack tip to have an appropriate interpolation of the damaged zone while for the rest of the solution domain standard shape functions are used to reduce the computational cost no new nodes are needed when defining the higher order shape functions by reproducing kernel element method and thus remeshing is avoided during the crack growth after describing the formulations several numerical examples are implemented to investigate the ability of the proposed method for both the linear elastic fracture mechanics and the gradient enhanced damage models

numerical algorithms and mesh dependence in the disturbed state concept the disturbed state concept dsc provides a unified approach for constitutive modelling of engineering materials including such factors as elastic plastic and creep strains microcracking damage and softening and sti ening responses the interacting mechanisms in the material mixture composed of the relative intact and fully adjusted states provide implicitly for various factors such as microcrack interaction and character istic dimension the dsc model can allow for well posedness reduction or elimination of spurious mesh dependence and localization a number of problems are solved to illustrate convergence and uniqueness of the finite element procedures localization spurious mesh dependence and validation with respect to observed behavior of simulated and practical problems 1997 by john wiley sons ltd

an equivalent stress gradient regularization model for coupled damage viscoelasticity abstract for materials that display viscoelastic behavior adequate description of their failure requires accurate prediction of damage initiation propagation and growth rate in addition to their time dependent response while many local damage models are available in the literature they all lack a length scale needed to regularize the solution and lead to mesh independent results in this work we propose a new damage regularization approach based on an equivalent stress measure concept and apply it to a prony series type viscoelastic solid with a murakami type damage rate law viscoelastic behavior is achieved by a semi analytical integration of the constitutive law and damage regularization is obtained by solving an additional second order gradient equation of an equivalent stress the scheme leads to a coupled set of nonlinear equations which are solved simultaneously using a monolithic newton framework to obtain displacement and damage fields as a function of time the formulation is shown to be thermodynamically consistent and the energy dissipation of the model is studied mesh insensitive behavior under creep relaxation and strain rate conditions is demonstrated for one and two dimensional problems moreover for different viscoelastic materials with different dominant damage terms the proposed model is shown to provide consistent damage growth results

an evaluation of higher order plasticity theories for predicting size effects and localisation conventional plasticity theories are unable to capture the observed increase in strength of metallic structures with diminishing size they also give rise to ill posed boundary value problems at the onset of material softening in order to overcome both deficiencies a range of higher order plasticity theories have been formulated in the literature the purpose of this paper is to compare existing higher order theories for the prediction of a size effect and the handling of localisation effects to this end size effect predictions for foils in bending are compared with existing experimental data furthermore a study of one dimensional harmonic incremental solutions from a uniform reference state allows one to assess the nature of material localisation as predicted by these competing higher order theories these analyses show that only one of the theories considered the fleck hutchinson strain gradient plasticity theory based upon the toupin mindlin strain gradient framework fleck n a hutchinson j w 1997 strain gradient plasticity adv appl mech 33 295 361 allows one to describe both phenomena the other theories show either nonphysical size effects or a pathologically localised post peak response

theory and numerics of a thermodynamically consistent framework for geometrically linear gradient plasticity the paper presents the theory and the numerics of a thermodynamically consistent formulation of gradient plasticity at small strains starting from the classical local continuum formulation which fails to produce physically meaningful and numerically converging results within localization computations a thermodynamically motivated gradient plasticity formulation is envisioned the model is based on an assumption for the helmholtz free energy incorporating the gradient of the internal history variable a yield condition and the postulate of maximum dissipation resulting in an associated structure as a result the driving force conjugated to the hardening evolution is identified as the quasi non local drag stress which incorporates besides the strictly local drag stress essentially the divergence of a vectorial hardening flux at the numerical side besides the balance of linear momentum the algorithmic consistency condition has to be solved in weak form thereby the crucial issue is the determination of the active constraints exhibiting plastic loading which is solved by an active set search algorithm borrowed from convex non linear programming moreover different discretization techniques are proposed in order to compare the fe performance in local plasticity with the advocated gradient formulation both for hardening and softening copyright 2001 john wiley sons ltd

continuum damage model for orthotropic materials application to masonry this paper contributes to the formulation of continuum damage models for orthotropic materials under plane stress conditions two stress transformation tensors related to tensile and compressive stress states respectively are used to establish a one to one mapping relationship between the orthotropic behaviour and an auxiliary model this allows the consideration of two individual damage criteria according to different failure mechanisms i e cracking and crushing the constitutive model adopted in the mapped space makes use of two scalar variables which monitor the local damage under tension and compression respectively the model affords the simulation of orthotropic induced damage while also accounting for unilateral effects thanks to a stress tensor split into tensile and compressive contributions the fundamentals of the method are presented together with the procedure utilized to adjust the model in order to study the mechanical behaviour of masonry material the validation of the model is carried out by means of comparisons with experimental results on different types of orthotropic masonry at the material level

positivity of line bundles on general blow ups of p2 abstract let x be the blow up of p 2 at r general points p 1 p r p 2 we study line bundles on x given by plane curves of degree d passing through p i with multiplicity at least m i motivated by results in 39 we establish conditions for ampleness very ampleness and global generation of such line bundles

finite strain logarithmic hyperelasto plasticity with softening a strongly non local implicit gradient framework this paper addresses the extension of a eulerian logarithmic finite strain hyperelasto plasticity model in order to incorporate an isotropic plastic damage variable that leads to softening and failure of the plastic material it is shown that a logarithmic elasto plastic model with a strongly non local degrading yield stress exactly preserves the structure of its infinitesimal counterpart the strongly non local nature of the model makes it an attractive framework for the numerical solution of softening plasticity problems consistent constitutive tangent operators are derived for the particular case of hyperelasto j2 plasticity which are exactly equal to the corresponding infinitesimal tangent operators the finite element implementation along with the geometrically nonlinear contributions and the incremental solution strategy is outlined a benchmark example is solved illustrating the main differences between the purely elasto plastic case and the case with plastic damage finally the main model characteristics and its practical use are emphasized

spectral analysis of localization in nonlocal and over nonlocal materials with softening plasticity or damage the paper shows that spectral wave propagation analysis reveals in a simple and clear manner the effectiveness of various regularization techniques for softening materials i e materials for which the yield limits soften as a function of the total strain both plasticity and damage models are considered it is verified analytically in a simple way that the nonlocal integral type model with degrading yield limit depending on the total strain works correctly if and only one adopts an unconventional nonlocal formulation introduced in 1994 by vermeer and brinkgreve and in 1996 by planas and by stromberg and ristinmaa which is here called for the sake of brevity over nonlocal because it uses a linear combination of local and nonlocal variables in which a negative weight imposed on the local variable is com pensated by assigning to the nonlocal variable weight greater than 1 this is equivalent to a nonlocal variable with a smooth positive weight function of total weight greater than 1 normalized by superposing a negative delta function spike at the center the spectral approach readily confirms that the nonlocal integral type generalization of softening plasticity with an additive format gives correct localization properties only if an over nonlocal formulation is adopted by contrast the nonlocal integral type generalization of softening plasticity with a multiplicative format provides real istic localization behavior just like the nonlocal integral type damage model and thus does not necessitate an over non local formulation the localization behavior of explicit and implicit gradient type models is also analyzed a simple analysis shows that plasticity and damage models with gradient type localization limiter whether explicit or implicit have very different localization behaviors 2005 elsevier ltd all rights reserved

stress based nonlocal damage model abstract progressive microcracking in brittle or quasi brittle materials as described by damage models presents a softening behavior that in turn requires the use of regularization methods in order to maintain objective results such regularization methods which describe interactions between points provide some general properties including objectivity and the non alteration of a uniform field as well as drawbacks damage initiation free boundary a modification of the nonlocal integral regularization method that takes the stress state into account is proposed in this contribution the orientation and intensity of nonlocal interactions are modified in accordance with the stress state the fundamental framework of the original nonlocal method has been retained making it possible to maintain the method s advantages the modification is introduced through the weight function which in this modified version depends not only on the distance between two points as for the original model but also on the stress state at the remote point the efficiency of this novel approach is illustrated using several examples the proposed modification improves the numerical solution of problems observed in numerical simulations involving regularization techniques damage initiation and propagation in mode i as well as shear band formation are analyzed herein

a geometrically regularized gradient damage model with energetic equivalence abstract aiming to bridge the gap between damage and fracture mechanics for modeling localized failure in solids this paper addresses a novel geometrically regularized gradient damage model with energetic equivalence for cracking evolution with the free energy potential defined as usual in terms of the local strain and damage fields the constitutive relations are derived consistently from the standard framework of thermodynamics upon the sharp crack topology geometrically regularized by the damage localization band with a length scale the ensuing energetic equivalence naturally yields the damage evolution law of gradient type and the associated boundary condition of neumann type compared to other gradient damage models no extra assumptions like the nonlocal energy residual and insulation condition are introduced moreover the damage gradient physically accounting for microscopic nonlocal interactions is fully dissipative as expected in line with the unified phase field theory recently proposed by the author wu 2017 during failure processes the material behavior is uniquely characterized by two constitutive functions i e the degradation function defining the free energy potential of the bulk and the geometric function regularizing the sharp crack topology in particular optimal constitutive functions defining an equivalent cohesive zone model of general softening laws are postulated with the involved parameters calibrated from standard material properties the proposed model is numerically implemented into the multi field finite element method and applied to several benchmark tests of concrete under mode i and mixed mode failure it is found that the incorporated length scale can be regarded either a numerical parameter or a material property for the former considered in this work the length scale has negligible if not no effects on the global responses so long as the sharp crack topology and the damage field of high gradients within the localization band are well resolved furthermore the localization bandwidth does not exhibit spurious widening but rather it approaches to a finite value proportional to the length scale comparison between the numerical and experimental results regarding the curve of load versus displacement and crack path confirms validity of the proposed gradient damage model for characterizing localized failure in quasi brittle solids

an inverse micro mechanical analysis toward the stochastic homogenization of nonlinear random composites abstract an inverse mean field homogenization mfh process is developed to improve the computational efficiency of non linear stochastic multiscale analyzes by relying on a micro mechanics model first full field simulations of composite stochastic volume element sve realizations are performed to characterize the homogenized stochastic behavior the uncertainties observed in the non linear homogenized response which result from the uncertainties of their micro structures are then translated to an incremental secant mfh formulation by defining the mfh input parameters as random effective properties these effective input parameters which correspond to the micro structure geometrical information and to the material phases model parameters are identified by conducting an inverse analysis from the full field homogenized responses compared to the direct finite element analyzes on sves the resulting stochastic mfh process reduces not only the computational cost but also the order of uncertain parameters in the composite micro structures leading to a stochastic mean field reduced order model mf rom a data driven stochastic model is then built in order to generate the random effective properties under the form of a random field used as entry for the stochastic mf rom embedded in a stochastic finite element method sfem the two cases of elastic unidirectional ud fibers embedded in an elasto plastic matrix and of elastic ud fibers embedded in a damage enhanced elasto plastic matrix are successively considered in order to illustrate the capabilities of the method the stochastic response of a ply is studied under transverse loading condition

strongly non local gradient enhanced finite strain elastoplasticity this paper presents a gradient plasticity formulation developed in a strongly non local format and applicable to ductile failure for finite strain plasticity the obtained formulation is an extension of a framework proposed by simo computer methods in applied mechanics and engineering 1988 66 199 into the softening and localization regime the presented model inherits the well established regularization properties of its strongly non local enrichment and leads to an implicit computational scheme with a consistent tangent operator numerical examples are given which illustrate the strongly non local character of the formulation the influence of the type of intrinsic length scale considered as well as several topics on plastic failure initiation and evolution into the geometrically non linear regime copyright 2003 john wiley sons ltd

existence uniqueness and uniform decay for the nonlinear beam degenerate equation with weak damping in this paper we prove global existence and uniqueness of weak solutions of the problem for the nonlinear beam degenerate equationk x t u email protected email protected w u 2dx email protected u email protected 0 whereq is a cylindrical domain of r n 1 n 1 with the lateral boundary s and k x t is a real function defined in q k x t 0 for all x t email protected 0 which satisfies some appropriate conditions m l is a real function such that m l email protected 0 b l 1 l 1 is the first eigenvalue of the operator d 2 moreover the uniform decay rates of the energy are obtained as time goes to infinity

extended embedded finite elements with continuous displacement jumps for the modeling of localized failure in solids abstract this paper addresses novel extended embedded finite elements with continuous displacement jumps for the modeling of localized failure in solids on one hand based upon the unified multiscale kinematics of strong discontinuities standard finite elements at the coarse scale are enriched with fine scale kinematics in which non uniform discontinuity modes are incorporated on the other hand the traction continuity condition across the discontinuity is enforced in the statically optimal form as the fine scale statics the admissible discontinuity modes satisfying a priori the traction continuity condition are derived in addition to the relative rigid body motions translations and rotations more general relative stretching modes that induce discontinuous strain stress fields in the bulk are obtained with no limit of a vanishing poisson s ratio the proposed model then particularizes to 2d quadrilateral elements the displacement jumps at the discontinuity nodes are selected as the enrichment parameters and regarded as global variables shared by neighboring elements the resulting model not only is intrinsically stress locking free for a fully softened discontinuity but also automatically guarantees global continuity of the displacement jumps another benefit is that the numerical instability suffered under the unfavorable element discontinuity configuration can be easily circumvented furthermore the vanishing displacement jumps at the discontinuity tip can be enforced trivially as conventional essential boundary conditions and those complex strategies introduced in existing models are avoided together with the failure criterion related to the element average stress and the propagation orientation based on a simplified nonlocal stress a local crack tracking algorithm is adopted to guarantee global continuity of the discontinuity path representative numerical simulations of element and structure benchmark tests are presented to verify the performance of the proposed approach in the modeling of arbitrary discontinuity propagation in solids

damage gradient of damage and principle of virtual power a theory of continuum damage mechanics is developed within the framework of the principle of virtual power because the damage in a solid results from microscopic movements we decided to include the power of these microscopic movements in the power of the internal forces the microscopic velocities are related to the damage rate the power of the internal forces we choose depends on the damage velocity and its gradient to take into account the interactions models issued from this theory are presented they are coherent from the mechanical and mathematical points of view the numerical computations show no mesh sensitivity they describe with good agreement the main experimental properties concrete is chosen as an example to illustrate the theory a model using two damage quantities is also presented it yields the correct description of the unilateral phenomenon observed in concrete finally an extension of the existing models is proposed to describe the fatigue damage behaviour

a phase field model for cohesive fracture in this paper a phase field model for cohesive fracture is developed after casting the cohesive zone approach in an energetic framework which is suitable for incorporation in phase field approaches the phase field approach to brittle fracture is recapitulated the approximation to the dirac function is discussed with particular emphasis on the dirichlet boundary conditions that arise in the phase field approximation the accuracy of the discretisation of the phase field including the sensitivity to the parameter that balances the field and the boundary contributions is assessed at the hand of a simple example the relation to gradient enhanced damage models is highlighted and some comments on the similarities and the differences between phase field approaches to fracture and gradient damage models are made a phase field representation for cohesive fracture is elaborated starting from the aforementioned energetic framework the strong as well as the weak formats are presented the latter being the starting point for the ensuing finite element discretisation which involves three fields the displacement field an auxiliary field that represents the jump in the displacement across the crack and the phase field compared to phase field approaches for brittle fracture the modelling of the jump of the displacement across the crack is a complication and the current work provides evidence that an additional constraint has to be provided in the sense that the auxiliary field must be constant in the direction orthogonal to the crack the sensitivity of the results with respect to the numerical parameter needed to enforce this constraint is investigated as well as how the results depend on the orders of the discretisation of the three fields finally examples are given that demonstrate grid insensitivity for adhesive and for cohesive failure the latter example being somewhat limited because only straight crack propagation is considered

decay rate estimates for the quasi linear timoshenko system with nonlinear control and damping terms in this work we consider a class of quasi linear timoshenko systems with nonlinear control and damping terms we investigate the stabilization of weak solutions to the initial boundary value problem and obtain explicit decay rate estimation depending on control and damping terms

existence uniqueness and uniform decay for the non linear degenerate equation with memory condition at the boundary this paper is concerned with the existence uniqueness and uniform decay for the solution to the non linear degenerate equation with memory condition at the boundary

powell sabin b splines for smeared and discrete approaches to fracture in quasi brittle materials abstract non uniform rational b splines nurbs and t splines can have some drawbacks when modelling damage and fracture the use of powell sabin b splines which are based on triangles can by pass these drawbacks herein smeared as well as discrete approaches to fracture in quasi brittle materials using powell sabin b splines are considered for the smeared formulation an implicit fourth order gradient damage model is adopted since quadratic powell sabin b splines employ c 1 continuous basis functions throughout the domain they are well suited for solving the fourth order partial differential equation that emerges in this higher order damage model moreover they can be generated from an arbitrary triangulation without user intervention since powell sabin b splines are generated from a classical triangulation they are not necessarily boundary fitting and in that case they are not isogeometric in the strict sense for discrete fracture approaches the degree of continuity of t splines is reduced to c 0 at the crack tip hence stresses need to be evaluated and weighted at the integration points in the vicinity of the crack tip in order to decide when the critical stress is reached in practice stress fields are highly irregular around crack tips furthermore aligning a t spline mesh with the new crack segment can be difficult powell sabin b splines also remedy these drawbacks as they are c 1 continuous at the crack tip and stresses can be directly computed which vastly increases the accuracy and simplifies the implementation moreover re meshing is more straightforward using powell sabin b splines a current limitation is that in three dimensions there is no procedure yet for constructing powell sabin b splines on arbitrary tetrahedral meshes

gradient dependent plasticity formulation and algorithmic aspects a plasticity theory is proposed in which the yield strength not only depends on an equivalent plastic strain measure hardening parameter but also on the laplacian thereof the consistency condition now results in a differential equation instead of an algebraic equation as in conventional plasticity to properly solve the set of non linear differential equations the plastic multiplier is discretized in addition to the usual discretization of the displacements for appropriate boundary conditions this formulation can also be derived from a variational principle accordingly the theory is complete

a critical comparison of nonlocal and gradient enhanced softening continua continuous models of material degradation may cease to produced meaningful results in the presence of high strain gradients these gradients may occur for instance in the propagation of waves with high wave numbers and at stress concentrators adding nonlocal or gradient terms to the constitutive modelling may enhance the ability of the models to describe such situations the effect of adding nonlocal or gradient terms and the relation between these enhancements are examined in a continuum damage setting a nonlocal damage model and two different gradient damage models are considered in one of the gradient models higher order deformation gradients enter the equilibrium equations explicitly while in the other model the gradient influence follows in a more implicit way from an additional partial differential equation the latter implicit gradient formulation can be rewritten in the integral format of the nonlocal model and can therefore be regarded as truly nonlocal this is not true for the explicit formulation in which the nonlocality is limited to an infinitesimal volume this fundamental difference between the formulations results in quite different behaviour in wave propagation localisation and at crack tips this is shown for the propagation of waves in the models their localisation properties and the behaviour at a crack tip the responses of the nonlocal model and the implicit gradient model agree remarkably well in these situations while the explicit gradient formulation shows an entirely different and sometimes nonphysical response

concurrent development of local and non local damage with the thick level set approach implementation aspects and application to quasi brittle failure abstract the present paper focuses on the discretization and implementation of the thick level set tls approach for quasi brittle failure when dealing with both local and non local development of damage the tls damage model introduces a length scale and nonlocality in the fracture process zone fpz and it handles the continuous to discontinuous failure transition the added nonlocality is limited to the fpz and therefore the model is local away from the fpz or before its emergence it leads to concurrent development of local and nonlocal damage during simulations this paper presents a space time discretization that handles concurrent computations in a unified manner it is based on an explicit time discretization written as a predictor corrector scheme and a space discretization that uses specific approximation functions modes that embed nonlocality it also uses an alternative implementation of the extended finite element method x fem enrichment used in the tls which is based on virtual nodes

unified analysis of enriched finite elements for modeling cohesive cracks abstract aiming to provide a justified theoretical ground based on which different enriched finite elements can be systematically compared we developed in this work a unified framework for modeling cohesive cracks using the variational multiscale method the kinematics i e coarse and fine scale displacement and strain fields and statics i e coarse and fine scale equilibrium equations are thoroughly investigated in both continuum and discrete settings with respect to the fine scale kinematics and statics adopted in the embedded finite element method efem and the extended finite element method xfem we mainly discuss four groups of enriched finite elements with non uniform discontinuity modes i e the kinematically optimal symmetric efem and xfem as well as the kinematically and statically optimal non symmetric efem and xfem in all these methods the enrichment parameters can be regarded either as element wise local or continuous global variables the enriched finite elements with material element node enrichments are then exemplified in an increasing order of their respective capabilities in representing the fine scale kinematics and the interrelations between them are also discussed owing to this unified framework we are able to clarify some existing points of view related to efem and xfem it is found that if the same fine scale statics is used and the same local global property is assumed for the involved enrichment parameters xfem can be regarded as a kinematically enhanced efem since it accounts for a more general fine scale kinematics than that i e relative rigid body motions and self stretching considered in efem finally several simple but sufficiently representative numerical examples are presented to show the significance in appropriately reflecting the fine scale kinematics and statics in an enriched finite element for modeling cohesive cracks

xfem based crack detection scheme using a genetic algorithm a new computational tool is developed for the accurate detection and identification of cracks in structures to be used in conjunction with non destructive testing of specimens it is based on the solution of an inverse problem based on some measurements typically along part of the boundary of the structure that describe the response of the structure to vibration in a chosen frequency or a combination of frequencies the goal is to estimate whether the structure contains a crack and if so to find the parameters location size orientation and shape of the crack that produces a response closest to the given measurement data in some chosen norm the inverse problem is solved using a genetic algorithm ga the ga optimization process requires the solution of a very large amount of forward problems the latter are solved via the extended finite element method xfem this enables one to employ the same regular mesh for all the forward problems performance of the method is demonstrated via a number of numerical examples involving a cracked flat membrane various computational aspects of the method are discussed including the a priori estimation of the ill posedness of the crack identification problem copyright 2007 john wiley sons ltd

a higher order stress based gradient enhanced damage model based on isogeometric analysis abstract the micro damage associated with diffuse fracture processes in quasi brittle materials can be described by continuum damage mechanics in order to overcome the mesh dependence of local damage formulations non local and gradient enhanced approaches are often employed in this manuscript a higher order stress based gradient enhanced formulation is proposed which exploits the higher order continuity of b spline functions in isogeometric analysis iga the proposed formulation does not require the decomposition of the fourth order model into two second order models two numerical examples are presented to demonstrate the performance of the formulation and to compare the obtained solutions with results from conventional gradient enhanced damage formulation

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

stabilized four node tetrahedron with nonlocal pressure for modeling hyperelastic materials summary non linear hyperelastic response of reinforced elastomers is modeled using a novel three dimensional mixed finite element method with a nonlocal pressure field the element is unconditionally convergent and free of spurious pressure modes nonlocal pressure is obtained by an implicit gradient technique and obeys the helmholtz equation physical motivation for this nonlocality is shown an implicit finite element scheme with consistent linearization is presented finally several hyperelastic examples are solved to demonstrate the computational algorithm including the inf sup and verifications tests copyright q 2008 john wiley sons ltd

from continuous to discontinuous failure in a gradient enhanced continuum damage model abstract a computational framework for the description of the combined continuous discontinuous failure in a regularised strain softening continuum is proposed the continuum is regularised through the introduction of gradient terms into the constitutive equations at the transition to discrete failure the problem fields are enhanced through a discontinuous interpolation based on the partition of unity paradigm of finite element shape functions the inclusion of internal discontinuity surfaces where boundary conditions are applied without modifications of the original finite element mesh avoids the unrealistic damage growth typical of this class of regularised continuum models combined models allow for the analysis of the entire failure process from diffuse microcracking to localised macrocracks the discretisation procedure is described in detail and numerical examples illustrate the performance of the combined continuous discontinuous approach

positivity of line bundles on general blow ups of abelian surfaces abstract let s l s be a polarized abelian surface and let m c l s i 1 r e i be a line bundle on bl r s where bl r s s is the blow up of s at r general points with exceptional divisors e 1 e r in this paper we provide a criterion for k very ampleness of m also we deal with the case when s is an arbitrary surface of picard number one with a numerically trivial canonical divisor

a level set based model for damage growth the thick level set approach in this paper we introduce a new way to model damage growth in solids a level set is used to separate the undamaged zone from the damaged zone in the damaged zone the damage variable is an explicit function of the level set this function is a parameter of the model beyond a critical length we assume the material to be totally damaged thus allowing a straightforward transition to fracture the damage growth is expressed as a level set propagation the configurational force driving the damage front is non local in the sense that it averages information over the thickness in the wake of the front the computational and theoretical advantages of the new damage model are stressed numerical examples demonstrate the capability of the new model to initiate cracks and propagate them even in complex topological patterns branching and merging for instance

on micro to macro connections in domain decomposition multiscale methods abstract micro to macro connection techniques constitute a key ingredient in the formulation of multiscale strategies in this contribution several connection methods are explored for the concurrent multiscale analysis of brittle heterogeneous materials particularly these techniques are investigated in a domain decomposition strong coupling multiscale framework the structural component under analysis is partitioned into a number of non overlapping domains and a fine scale resolution is assigned therein in an adaptive manner exploiting a zoom in technique mesh refinement is employed in the domains where crack coalescence and growth take place the original contribution presented in this manuscript consists in the study of different strong and weak locality constraints that connect coarse and fine resolution domains standard collocation and average compatibility are considered and serve as a basis for the development of two new interscale links the influence of different locality constraints is studied in terms of the mechanical response and the error distribution is compared to a full fine scale analysis

a discontinuous galerkin method for strain gradient dependent damage study of interpolations and convergence abstract a discontinuous galerkin method has been developed for strain gradient dependent damage the strength of this method lies in the fact that it allows the use of c 0 interpolation functions for continuum theories involving higher order derivatives while in a conventional framework at least c 1 interpolations are required the discontinuous galerkin formulation thereby offers significant potential for engineering computations with strain gradient dependent models when using basis functions with a low degree of continuity jump conditions arise at element edges which are incorporated in the weak form in addition to the formulation itself a detailed study of the convergence properties of the method for various element types is presented and an error analysis is undertaken numerical results of some one and two dimensional problems are presented and discussed