discontinuous displacement approximation for capturing plastic localization it is proposed to capture localized plastic deformation via the inclusion of regularized displacement discontinuities at element boundaries interfaces of the finite element subdivision the regularization is based on a kinematic assumption for an interface that resembles that which is pertinent to the classical shear band concept as a by product of the regularization an intrinsic band width is introduced as a constitutive property rather than a geometric feature of the finite element mesh in this way the spurious mesh sensitivity which is obtained when the displacement approximation is continuous can be avoided another consequence is that the interfacial relation between the elements is derived directly from the conventional constitutive properties of the continuously deforming material an interesting feature is that the acoustic tensor will not only play a role for diagnosing discontinuous bifurcation but will also serve as the tangent stiffness tensor of the interface up to within a scalar factor an analytical investigation of the behaviour of the interface is carried out and it is shown that dilatation may indeed accompany slip within a shear band for a general plasticity model the significance of proper mesh alignment is demonstrated for a simple problem in plane strain and plane stress it is shown that a unique structural post peak response in accordance with non linear fracture mechanics can be achieved when the plastic softening modulus is properly related to the bandwidth the paper concludes with a numerical simulation of the gradual development of a shear band in a soil slope

localisation in a cosserat continuum under static and dynamic loading conditions abstract localisation studies have been carried out for an unconstrained elasto plastic strain softening cosserat continuum because of the presence of an internal length scale in this continuum model a perfect convergence is found upon mesh refinement a finite constant width of the localisation zone and a finite energy dissipation are computed under static as well as under transient loading conditions because of the existence of rotational degrees of freedom in a cosserat continuum additional wave types arise and wave propagation becomes dispersive this has been investigated analytically and numerically for an elastic cosserat continuum and an excellent agreement has been found between both solutions

embedded crack vs smeared crack models a comparison of elementwise discontinuous crack path approaches with emphasis on mesh bias this paper is concerned with an analysis of the mesh dependence of results from numerical failure analyses using the strong discontinuity approach sda based on an analytical and numerical comparison with a standard plasticity model pertaining to the classical fracture energy based smeared crack concept using a consistent characteristic length the sda is characterized by incorporating the kinematics of fractured solids locally within the finite element the resulting jumps in the displacement field are captured by their corresponding singularly distributed strains for the analytical and numerical study of the mesh dependence only mode i failure is considered analogies and differences of both finite element formulations are investigated and evaluated

a novel algorithmic framework for the numerical implementation of locally embedded strong discontinuities in this paper a novel algorithmic framework for the numerical modeling of locally embedded strong discontinuities suitable for the analysis of material failure such as cracking in brittle structures or shear bands in soils is proposed based on the enhanced assumed strain eas concept the final failure kinematics of solids approximated by discontinuous displacement fields are incorporated into the finite element formulation in contrast to previous works the discontinuous part of the deformation mapping is condensed out at the material level without employing the standard static condensation technique as a consequence the resulting constitutive equations are formally identical to those of standard plasticity models hence the return mapping algorithm is applied to the numerical implementation only slight modifications of this by now classical algorithm are necessary the proposed framework is applicable to any constitutive equation characterizing the inelastic part of deformation referring to the yield failure function and the evolution equations no special assumption has to be made despite the differences between the presented numerical framework and the original implementation of the strong discontinuity approach sda it is shown that both finite element models are completely equivalent the applicability of the novel algorithmic formulation and its numerical performance are investigated by means of two two dimensional as well as by means of a fully three dimensional numerical analysis of shear band formation

a relaxation approach applied to domain optimization a domain optimization problems related to potential theory is studied by means of a relaxation approach where a concave functional on a given convex set is defined the functional has a minimizing point that is the characteristic function of an optimal domain as a consequence of the necessary conditions of optimality the domain is the solution of a free boundary value problem

towards variational constitutive updates for non associative plasticity models at finite strain models based on a volumetric deviatoric split in this paper an enhanced variational constitutive update suitable for a class of non associative plasticity theories at finite strain is proposed in line with classical numerical formulations for plasticity models such as the by now established return mapping algorithm variational constitutive updates represent a numerical method for computing the unknown state variables however in contrast to conventional algorithms variational constitutive updates are fully variational i e all unknown variables follow jointly from minimizing a certain potential in addition to the physical and mathematical elegance of these variational schemes they show several practical advantages as well for instance numerically efficient and robust optimization schemes can be directly employed for solving the resulting minimization problem since mathematically plasticity is a non smooth problem and often it leads to highly singular systems of equations as known from single crystal plasticity a robust implementation is of utmost importance so far variational constitutive updates have been developed for different classes of standard dissipative solids i e solids characterized by associative evolution equations and flow rules in the present paper this framework is extended to a certain class of non associative plasticity models at finite strain all models falling into this class show a volumetric deviatoric split of the helmholtz energy and the yield function typical prototypes are drucker prager or mohr coulomb models playing an important role in soil mechanics the efficiency and robustness of the resulting algorithmic formulation is demonstrated by means of selected numerical examples

enrichment of enhanced assumed strain approximations for representing strong discontinuities addressing volumetric incompressibility and the discontinuous patch test summary we present a geometrically nonlinear assumed strain method that allows for the presence of arbitrary intra finite element discontinuities in the deformation map special attention is placed on the coarsemesh accuracy of these methods and their ability to avoid mesh locking in the incompressible limit given an underlying mesh and an arbitrary failure surface we first construct an enriched approximation for the deformation map with the nonlinear analogue of the extended finite element method x fem with regard to the richer space of functions spanned by the gradient of the enriched approximation we then adopt a broader interpretation of variational consistency for the construction of the enhanced strain in particular in those elements intersected by the failure surface we construct enhanced strain approximations which are orthogonal to piecewise constant stress fields contrast is drawn with existing strong discontinuity approaches where the enhanced strain variations in localized elements were constructed to be orthogonal to constant nominal stress fields importantly the present formulation gives rise to a symmetric tangent stiffness matrix even in localized elements the present modification also allows for the satisfaction of a discontinuous patch test wherein two different constant stress fields on each side of the failure surface lie in the solution space we demonstrate how the proposed modifications eliminate spurious stress oscillations along the failure surface particularly for nearly incompressible material response additional numerical examples are provided to illustrate the efficacy of the modified method for problems in hyperelastic fracture mechanics copyright c 2000 john wiley sons ltd

three dimensional finite elements with embedded strong discontinuities to model material failure in the infinitesimal range summary this paper presents new three dimensional finite elements with embedded strong discontinuities in the small strain infinitesimal range the goal is to model localized surfaces of failure in solids such as cracks at fracture through enhancements of the finite elements that capture the propagating discontinuities of the displacement field in the element interiors in this way such surfaces of discontinuity can be sharply resolved in general meshes not necessarily related to the detailed geometry of the surface unknown a priori an important issue is also the consideration of general finite element formulations in the developments e g basic displacement based mixed or enhanced assumed strain finite element formulations as needed to optimally resolve the continuum problem in the bulk the actual modeling of the discontinuity effects including the incorporation of the cohesive law defining the discontinuity constitutive response is carried out at the element level with the proper enhancement of the discrete strain field of the element the added elemental degrees of freedom approximate the displacement jumps associated with the discontinuity and are defined independently from element to element thus allowing their static condensation at the element level without affecting the global mechanical problem in terms of the number and topology of the global degrees of freedom in fact this global local structure of the finite element methods developed in this work arises naturally from a multi scale characterization of these localized solutions with the discontinuities understood to appear in the small scales thus leading directly to these computationally efficient numerical methods for their numerical resolution easily incorporated to an existing finite element code the focus in this paper is on the development of finite elements incorporating a linear interpolation of the displacement jumps in the general three dimensional setting these interpolations are shown to be necessary for hexahedral elements to avoid the so called stress locking that occurs with simpler constant approximations of the jumps namely a spurious transfer of stresses across the discontinuity not allowing its full release and hence resulting in an overstiff or locked numerical solution the design of the new finite elements is accomplished in this work by a direct identification of the separation modes to be incorporated in the discrete strain field of the element rather than from an assumed discontinuous interpolation of the displacements assuring with this approach their locking free response by design an additional issue addressed in the paper is the geometric characterization and propagation of the discontinuity surfaces in the general three dimensional setting of interest here the paper includes a series of numerical simulations illustrating and evaluating the properties of the new finite elements copyright 2012 john wiley sons ltd

an algorithm based on incompatible modes for the global tracking of strong discontinuities in shear localization analyses abstract numerical methods for predicting localized shear failure in elasto plastic solids have experienced considerable advancements in the last decades among these approaches the so called embedded strong discontinuity esd method is often successfully used to accurately simulate the post localization response with negligible dependence on the finite element discretization however it was observed that the employed discontinuity tracking strategy plays a crucial role in the successful localization analysis in this contribution we propose a novel strategy for the global tracking of discontinuity surfaces it is based on exploiting information obtained from the enhanced parameters employed in enhanced assumed strain eas formulations it is well known that enhanced strain element formulations are able to better capture localized shear deformations as compared to standard finite elements this can be explained as a consequence of the improved performance in bending we observed that the approximation of the strain jumps delimiting the shear band is connected with a deformation field characterized by opposite bending curvatures across these two discontinuities hence in view of the relations existing between the kinematics of strong and weak discontinuities we formulate a proper scalar function of the enhanced parameters to identify potential strong discontinuity surfaces which are evaluated in each step of the analysis with negligible computational cost this proposed approach has a global character as it is based upon evaluating discontinuity surfaces defined in the complete analysis domain that are by construction continuous across elements we demonstrate that the tracking algorithm correctly identifies the potential strong discontinuity surface already in early loading stages even before a localization condition is fulfilled in those elements which are crossed by the potential failure surface and which also satisfy the localization condition the kinematics of embedded strong discontinuities is activated to capture the shear failure surface the performance of the new tracking algorithm is demonstrated by means of several numerical shear localization analyses using associative and non associative drucker prager elastoplastic models to simulate 2 d and 3 d benchmarkanalyses

stability and convergence proofs for a discontinuous galerkin based extended finite element method for fracture mechanics abstract we prove the optimal convergence of a discontinuous galerkin based extended finite element method for two dimensional linear elastostatic problems over cracked domains the method which we proposed earlier 1 has two distinctive traits a it enriches the finite element space with the modes i and ii singular asymptotic crack tip fields over a neighborhood of the crack tip termed the enrichment region and b it allows functions in the finite element space to be discontinuous across the boundary between the enrichment region and the rest of the domain the treatment for this discontinuity generally a non polynomial function is facilitated by a specially designed discontinuous galerkin method based on the bassi rebay numerical flux the stability of the method is contingent upon an inf sup condition which we have proved to hold for any quasiuniform mesh family with sufficiently fine meshes we have also shown the optimal convergence of the displacement and stress fields and the convergence of the stress intensity factors extracted as the coefficients of the enrichment functions

an embedded strong discontinuity model for cracking of plain concrete abstract a numerical model formulated within the framework of a nonsymmetric strong discontinuity approach for fracture simulations of plain concrete is presented the model is based on the fixed crack concept and makes use of the concept of the elements with embedded discontinuities discontinuity segments of individual elements are considered to form a c0 continuous path enforcement of continuity of the crack path across adjacent elements is established by a partial domain crack tracking algorithm orientation of individual crack segments is derived from a nonlocal strain field the capabilities of the model are shown by means of numerical examples

assumed enhanced strain and the extended finite element methods a unification of concepts abstract we present strong and weak forms of boundary value problems for a solid with a cohesive frictional crack two commonly used techniques for enhancing the finite element fe interpolation are considered the assumed enhanced strain aes and the extended fe methods we compare and contrast the fe approximations involved in these techniques the aes method employs a piecewise constant interpolation of slip that is discontinuous across element boundaries in contrast the extended fe method interpolates slip continuously across element boundaries through numerical examples we discuss the implications of these approximations to the calculated overall deformation field in the regime of infinitesimal deformation with only a heaviside enrichment we find the aes method to predict larger slip i e softer response compared to the extended fe solution we ascribe the discrepancy to the lack of higher order crack tip enhancement in the extended fe solution finally we compare the slip predicted by the above fe enhancement techniques to that calculated by classical nonlinear contact mechanics algorithm for the case where the crack traverses the element sides

new finite elements with embedded strong discontinuities in the finite deformation range this paper develops new finite elements with embedded strong discontinuities for the modeling of the failure of solids in the finite deformation range the cases of interest include cracks shear bands delamination in composites and similar failures modeled by a propagating discontinuity in the deformation mapping that incorporates a cohesive law in terms of the associated displacement jumps the new elements accommodate the discontinuity in their interiors with a linear interpolation of the displacement jumps in contrast to the existing elements with constant jumps a general methodology is proposed for the development of such finite elements but the final focus is on quadrilateral elements for plane problems where a constant jump interpolation leads to a spurious transfer of stresses through the discontinuity or stress locking in certain common configurations the key aspect of the new elements is the proper enhancement of the deformation gradient that can represent correctly the kinematics of linearly separating discontinuities including a relative finite rotation across the discontinuity and in a frame indifferent manner the enhanced parameters modeling the displacement jumps are introduced independently for each element allowing for their static condensation at the element level and hence to arrive at a global system of equations with the same number and connectivity of the original global degrees of freedom for the underlying finite element mesh this situation leads to a particularly simple implementation of the proposed approach in an existing finite element code the performance of the new elements is illustrated with several numerical simulations including delamination in composite materials and propagating shear bands in a finitely deforming elastoplastic solid

a contact algorithm for frictional crack propagation with the extended finite element method we present an incremental quasi static contact algorithm for path dependent frictional crack propagation in the framework of the extended finite element fe method the discrete formulation allows for the modeling of frictional contact independent of the fe mesh standard coulomb plasticity model is introduced to model the frictional contact on the surface of discontinuity the contact constraint is borrowed from non linear contact mechanics and embedded within a localized element by penalty method newton raphson iteration with consistent linearization is used to advance the solution we show the superior convergence performance of the proposed iterative method compared with a previously published algorithm called latin for frictional crack propagation numerical examples include simulation of crack initiation and propagation in 2d plane strain with and without bulk plasticity in the presence of bulk plasticity the problem is also solved using an augmented lagrangian procedure to demonstrate the efficacy and adequacy of the standard penalty solution copyright 2008 john wiley sons ltd

embedded localization band in undrained soil based on regularized strong discontinuity theory and fe analysis abstract the paper presents a novel approach to the analysis of a developing localization zone in undrained soil considered as a mixture of a solid skeleton and fluid filled pores where the solid phase is considered as elastic plastic the basic feature is the concept of regularized displacement and pore pressure discontinuities which are assumed to occur across an internal discontinuity surface in this respect the present developments are extensions of those suggested by simo et al 1993 comput mech 12 277 296 and more recently by larsson and runesson 1995 j engng mech asce in press a finite element formulation is proposed on the basis of a mixed variational formulation in the spirit of the enhanced strain concept by simo and rifai 1990 int j numer meth engng 29 1595 1638 in this fashion the localization zone is embedded into a base element for the ordinary analysis of the non localized mixture problem

strain localization in frictional materials exhibiting displacement jumps abstract this paper presents a mathematical model for analyzing strain localization in frictional solids exhibiting displacement jumps precise conditions for the appearance of slip lines including their initiation and evolution are outlined for a rate independent strain softening drucker prager model and explicit analytical expressions are used to describe the orientation of the slip line a stress displacement relation obtained through the consistency condition is also formulated to describe the quasi static response in the post localization regime the mathematical model which is cast within the framework of finite element analysis employing the assumed enhanced strain method circumvents mesh dependence issues often associated with rate independent plasticity models it is shown that the enhancement equation is nothing else but the consistency condition imposed on the band numerical examples involving plane strain compression are described to demonstrate objectivity with respect to mesh refinement and insensitivity to mesh alignment of finite element solutions

on the mechanical energy and effective stress in saturated and unsaturated porous continua abstract this paper has two objectives a to formulate a mechanical theory of porous continua within the framework of strong discontinuity concept commonly employed for the analysis of strain localization in inelastic solids and b to introduce an effective stress tensor for three phase solid water air partially saturated porous continua emerging from principles of thermodynamics to achieve the first objective strong forms of the boundary value problems are compared between two formulations the first in which the velocity jump at the solid fluid interface is treated as a strong discontinuity problem and the second in which the strong discontinuity is smeared in the representative volume element as for the second objective an effective cauchy stress tensor of the form 1 k k s p 1 emerges from the formulation where is the total stress tensor k and ks are the bulk moduli of the solid matrix and solid phase respectively and p is the mean pore water and pore air pressures weighted according to the degree of saturation we show from the first and second laws of thermodynamics that this effective stress tensor is power conjugate to the solid rate of deformation tensor and that it includes the mechanical power required to compress the solid phase

a new generalized finite element method for two scale simulations of propagating cohesive fractures in 3 d summary this paper presents a novel numerical framework based on the generalized finite element method with global local enrichments gfemgl for two scale simulations of propagating fractures in three dimensions a non linear cohesive law is adopted to capture objectively the dissipated energy during the process of material degradation without the need of adaptive remeshing at the macro scale or artificial regularization parameters the cohesive crack is capable of propagating through the interior of finite elements in virtue of the partition of unity concept provided by the generalized extended finite element method and thus eliminating the need of interfacial surface elements to represent the geometry of discontinuities and the requirement of finite element meshes fitting the cohesive crack surface the proposed method employs fine scale solutions of non linear local boundary value problems extracted from the original global problem in order to not only construct scale bridging enrichment functions but also to identify damaged states in the global problem thus enabling accurate global solutions on coarse meshes this is in contrast with the available gfemgl in which the local solution field contributes only to the kinematic description of global solutions the robustness efficiency and accuracy of this approach are demonstrated by results obtained from representative numerical examples copyright 2015 john wiley sons ltd

aes for multiscale localization modeling in granular media this work presents a multiscale strong discontinuity approach to tackle key challenges in modeling localization behavior in granular media accommodation of discontinuities in the kinematic fields and direct linkage to the underlying grain scale information assumed enhanced strain aes concepts are borrowed to enhance elements for post localization analysis but are reformulated within a recently proposed hierarchical multiscale computational framework unlike classical aes methods where material properties are usually constants or assumed to evolve with some arbitrary phenomenological laws this framework provides a bridge to extract evolutions of key material parameters such as friction and dilatancy based on grain scale computational or experimental data more importantly the phenomenological softening modulus typically used in aes methods is no longer required numerical examples of plane strain compression tests are presented to illustrate the applicability of this method and to analyze its numerical performance

three dimensional finite elements with embedded strong discontinuities for the analysis of solids at failure in the finite deformation range abstract this paper presents the development of new three dimensional finite elements in the finite deformation range for the numerical modeling of material failure characterized by propagating crack type surfaces in solids commonly referred to as strong discontinuities the singular fields associated with such discontinuous solutions are captured by an element based enhancement locally within a multi scale framework the complex kinematics arising from the general propagating discontinuity surfaces especially in the considered finite deformation setting is captured by the direct identification of the element strain fields associated to the separation modes of the discontinuity rather than attempting to interpolate directly the corresponding discontinuous deformation fields to avoid the over stiff response known as stress locking by which the discontinuities cannot open as needed by the material response the new brick elements consider full linear interpolations of the displacement jumps this requires a total of nine local parameters for each localized finite element parameters that are condensed out at the element level the resulting formulation does not change then neither the number nor the connectivity of the global degrees of freedom employed in the finite element resolution of the underlying global mechanical problem thus leading to a computationally efficient numerical technique that captures sharply these discontinuous solutions in particular the localized dissipative mechanism associated to the material failure is captured objectively through the consideration of the proper cohesive law along the discontinuity surface special care is taken to assure the material frame indifference of the final formulation complete details of the implementation of the new elements including the consistent linearization of the governing equations are also presented the results obtained in numerical simulations of representative benchmark problems illustrate the performance of the new finite elements with embedded strong discontinuities

finite element methods for the analysis of strong discontinuities in coupled poro plastic media this paper presents the formulation of finite element methods for the numerical resolution of strong discontinuities in poro plastic solids fully coupled infinitesimal conditions are considered these solutions are characterized by a discontinuous displacement field with the associated singular strains and a singular distribution of the fluid content here singular distributions refer to dirac delta functions the singular component of the fluid content distribution models the fluid accumulated per unit area of the discontinuity surface and it is directly related with the dilatancy characterizing singular inelastic strains localized along such a surface it further accounts for a discontinuous fluid flow vector given by darcy s law in terms of a continuous pore pressure field all these considerations are incorporated in the proposed finite element methods through a local enhancement of the finite element interpolations as these discontinuities appear the local character of these interpolations lead after the static condensation of the enhanced fields to a large scale problem exhibiting the same structure as common finite element models of the global poro plastic problem but incorporating now crucially the localized dissipative effects characteristic of the localized failures several numerical simulations are presented to evaluate the performance of the proposed methods

finite element simulation of strain localization with large deformation capturing strong discontinuity using a petrov galerkin multiscale formulation the finite element model for strain localization analysis developed in a previous work is generalized to the finite deformation regime strain enhancements via jumps in the displacement field are captured and condensed on the material level leading to a formulation that does not require static condensation to be performed on the element level a general evolution condition is first formulated laying out conditions for the continued activation of a strong discontinuity then a multiscale finite element model is formulated to describe the post bifurcation behavior highlighting the key roles played by the continuous and conforming deformation maps on the characterization of the finite deformation kinematics of a localized element traced by a strong discontinuity the resulting finite element equation exhibits the features of a petrov galerkin formulation in which the gradient of the weighting function is evaluated over the continuous part of the deformation map whereas the gradient of the trial function is evaluated over the conforming part of the same map in the limit of infinitesimal deformations the formulation reduces to the standard galerkin approximation described in a previous work numerical examples are presented demonstrating absolute objectivity with respect to mesh refinement and insensitivity to mesh alignment of finite element solutions 2002 elsevier science b v all rights reserved

stabilized low order finite elements for frictional contact with the extended finite element method abstract contact problem suffers from a numerical instability similar to that encountered in incompressible elasticity in which the normal contact pressure exhibits spurious oscillation this oscillation does not go away with mesh refinement and in some cases it even gets worse as the mesh is refined using a lagrange multipliers formulation we trace this problem to non satisfaction of the lbb condition associated with equal order interpolation of slip and normal component of traction in this paper we employ a stabilized finite element formulation based on the polynomial pressure projection ppp technique which was used successfully for stokes equation and for coupled solid deformation fluid diffusion using low order mixed finite elements for the frictional contact problem the polynomial pressure projection approach is applied to the normal contact pressure in the framework of the extended finite element method we use low order linear triangular elements tetrahedral elements for 3d for both slip and normal pressure degrees of freedom and show the efficacy of the stabilized formulation on a variety of plane strain plane stress and three dimensional problems

a finite element model for strain localization analysis of strongly discontinuous fields based on standard galerkin approximation abstract this paper presents a finite element model for strain localization analysis of elastoplastic solids subjected to discontinuous displacement fields based on standard galerkin approximation strain enhancements via jumps in the displacement field are captured and condensed on the material level leading to a formulation that does not require static condensation to be performed on the element level the mathematical formulation revolves around the dual response of a macroscopic point cut by a shear band which requires the satisfaction of the yield condition on the band as the same stress point unloads elastically just outside the band precise conditions for the appearance of slip lines including their initiation and evolution are outlined for a rate independent strain softening drucker prager model and explicit analytical expressions are used to describe the orientation of the slip line in a plane strain setting at post localization the stress point integration algorithm along the band is exact and amenable to consistent linearization numerical examples involving simple shearing of elastoplastic solids with deviatoric plastic flow as well as plane strain compression of dilatant cohesive frictional materials are presented to demonstrate absolute objectivity with respect to mesh refinement and insensitivity to mesh alignment of finite element solutions

internal symmetry groups for the drucker prager material model of plasticity and numerical integrating methods abstract internal symmetries for the drucker prager material model and numerical methods based on them are investigated here we first convert the non linear constitutive equations to a lie type system x a x t x where a sl 5 2 r is a lie algebra of the special orthochronous pseudo linear group sl 5 2 r the underlying space of drucker prager model in the plastic phase is a pseudo sphere in the pseudo riemann manifold which is locally a pseudo euclidean space e 7 5 2 and whose metric tensor is indefinite having signature 5 2 and also depends on the temporal component then we split the drucker prager yield condition into two sub yield conditions which together with two integrating factors idea led us to derive two lie type systems in the product space m 5 1 m 1 1 the lie algebra is the direct sum so 5 1 so 1 1 and the symmetry group is thus soo 5 1 soo 1 1 these results are essential from computational aspect based on these symmetry groups exponential mappings are developed which update stress points exactly on the yield surface at every time increment without any iteration as an example the results calculated by using the two different group preserving schemes are compared to that calculated by the runge kutta scheme together with the newton raphson iterative solution of a non linear algebraic equation which enforcing the consistency condition

plane strain finite element analysis of pressure sensitive plasticity with strong discontinuity numerical simulations of localized deformation in solids should capture the structural phenomenon of localization and associated loss of material body strength in a manner independent of spatial discretization many regularization techniques have been proposed to address the ill posedness associated with rate independent softening plasticity that leads to mesh dependent numerical simulations one approach to alleviating mesh dependence is the strong discontinuity approach which represents localized deformation as a slip surface within a plasticity model a strong discontinuity is a discontinuous displacement field this approach is used in this paper to formulate a plane strain pressure sensitive nonassociative plasticity model with strong discontinuity and to implement the model along with an enhanced bilinear quadrilateral element via an assumed enhanced strain method numerical examples demonstrate mesh independence of the method for pressure sensitive materials

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

cam clay plasticity part iii extension of the infinitesimal model to include finite strains the infinitesimal version of the modified cam clay model of critical state soil mechanics is reformulated to include finite deformation effects central to the formulation are the choice of a hardening law that is appropriate for cases involving large plastic volumetric strains and the use of a class of two invariant stored energy functions appropriate for cam clay type models that includes as special cases the constant elastic shear modulus approximation as well as the pressure dependent shear modulus elastic model the analytical model is cast within the framework of finite deformation theory based on a multiplicative decomposition of the deformation gradient for the fully elasto plastic case return mapping is done implicitly in the space defined by the invariants of the elastic logarithmic principal stretches which requires the solution at each stress point of no more than three simultaneous non linear equations a finite element analysis of a strip footing problem is presented to illustrate a prototype example where finite deformation effects significantly impact the predicted response

exact gaussian beam theory for outgoing and standing spherical waves application to transmitting and receiving antennas outgoing spherical vector wave functions are expressed in terms of gaussian beams also called complex source point beams radiating in all directions by use of a vector wave expansion the electromagnetic field of an arbitrary source of finite extent is thus expressed in terms of gaussian beams with weights determined directly from the spherical expansion coefficients of the source these outgoing wave formulas allow the field of any transmitting antenna to be expressed in terms of gaussian beams elementary gaussian beam receivers are introduced as the electromagnetic field at complex points in space the outputs of elementary gaussian beam receivers pointing in all directions determine the spherical expansion coefficients in a standing wave expansion these standing wave formulas allow the output of any receiving antenna to be expressed in terms of the outputs of elementary gaussian beam receivers combining the formulas for outgoing and standing waves produces a new antenna antenna transmission formula based solely on gaussian beams the theory is exact and validated through numerical examples