a gradient enhanced large deformation continuum damage model for fibre reinforced materials a non local gradient based damage formulation within a geometrically non linear setting is presented the hyperelastic constitutive response at local material point level is governed by a strain energy which is additively composed of an isotropic matrix and of an anisotropic fibre reinforced material respectively the inelastic constitutive response is governed by a scalar 1 d type damage formulation where only the anisotropic elastic part is assumed to be affected by the damage following the concept in dimitrijevic and hackl 28 the local free energy function is enhanced by a gradient term this term essentially contains the gradient of the non local damage variable which itself is introduced as an additional independent variable in order to guarantee the equivalence between the local and non local damage variable a penalisation term is incorporated within the free energy function based on the principle of minimum total potential energy a coupled system of euler lagrange equations i e the balance of linear momentum and the balance of the non local damage field is obtained and solved in weak form the resulting coupled highly non linear system of equations is symmetric and can conveniently be solved by a standard incremental iterative newton raphson type solution scheme several three dimensional displacement and force driven boundary value problems partially motivated by biomechanical application highlight the mesh objective characteristics and constitutive properties of the model and illustratively underline the capabilities of the formulation proposed

on finite strain micromorphic elastoplasticity abstract in the micromorphic continuum theory of eringen it was proposed that microstructure of materials could be represented in a continuum framework using a micro deformation tensor governing micro element deformation in addition to the deformation gradient governing macro element deformation the paper formulates finite strain micromorphic elastoplasticity based on micromorphic continuum mechanics in the sense of eringen multiplicative decomposition into elastic and plastic parts of the deformation gradient and micro deformation are assumed and the clausius duhem inequality is formulated in the intermediate configuration b to analyze what stresses elastic deformation measures and plastic deformation rates are used defined in the constitutive equations the resulting forms of plastic and internal state variable evolution equations can be viewed as phenomenological at their various scales i e micro continuum and macro continuum the phenomenology of inelastic mechanical material response at the various scales can be different but for demonstration purposes j2 flow plasticity is assumed for each of three levels of plastic evolution equations identified with different stress internal state variables and material parameters all evolution equations and a semi implicit time integration scheme are formulated in the intermediate configuration for future coupled lagrangian finite element implementation a simpler two dimensional model for anti plane shear kinematics is formulated to demonstrate more clearly how such model equations simplify for future finite element implementation

a valued semicircular systems abstract to a von neurnann algebra a and a set of linear maps ij a a i j i such that a ij ij i a a b l2 i is normal and completely positive we associate a von neumann algebra a this von neumann algebra is generated by a and an a valued semicircular system xi i i associated to in many cases there is a faithful conditional expectation e a a if a is tracial then under certain assumptions on a also has a trace one can think of the construction a as an analogue of a crossed product construction we show that most known algebras arising in free probability theory can be obtained from the complex field by iterating the construction of a particular interest are free krieger algebras which by analogy with crossed products and ordinary krieger factors are defined to be algebras of the form l 0 1 the cores of free araki woods factors are free krieger algebras we study the free krieger algebras and as a result obtain several non isomorphism results for free araki woods factors as another source of classification results for free araki woods factors we compute the invariant of connes for free products of von neumann algebras this computation generalizes earlier work on computation of t s and sd invariants for free product algebras

some links between recent gradient thermo elasto plasticity theories and the thermomechanics of generalized continua this work elaborates upon two robust models of gradient elasticity and gradient plasticity and one gradient model of heat transfer as originally advocated by the second author in the 1980 s the objective is after recalling the links between these models and existing generalized continuum theories as developed in the 1960 s and subsequently to apply the same methodology to the case of diffusion with a view to establishing generalized transport equations aifantis double diffusivity and conductivity theory that provides generalized mass or heat transfer equations is compared to micromorphic type hyper temperature and micro entropy proposals the double temperature and the micromorphic thermal models are shown to lead to equations more general that cattaneo s the sign of the coefficient of the second time derivative of temperature is found to differ according to both approaches the double temperature model contains a fourth space derivative term not present in the micromorphic models such generalized equations can be useful for example in the interpretation of recent femtosecond laser experiments on metals

micromorphic approach for finite gradient elastoplasticity fully coupled with ductile damage formulation and computational aspects abstract it is well established that the use of inelastic constitutive equations accounting for induced softening leads to pathological space mesh and time discretization dependency of the numerical solution of the associated initial and boundary value problem ibvp to avoid this drawback many less or more approximate solutions have been proposed in the literature in order to regularize the ibvp and to obtain numerical solutions which are at convergence much less sensitive to the space and the time discretization the basic idea behind these regularization techniques is the formulation of nonlocal constitutive equations by introducing some effects of characteristic lengths representing the materials microstructure in this work using the framework of generalized nonlocal continua a thermodynamically consistent micromorphic formulation using appropriate micromorphic state variables and their first gradients is proposed in order to extend the classical local constitutive equations by incorporating appropriate characteristic internal lengths the isotropic damage the isotropic and the kinematic hardenings are supposed to carry the targeted micromorphic effects first the theoretical aspects of this fully coupled micromorphic formulation is presented in details and the proposed generalized balance equations as well as the fully coupled micromorphic constitutive equations deduced the associated numerical aspects in the framework of the classical galerkin based fe formulation are briefly discussed in the special case of micromorphic damage specifically the formulation of 2d finite elements with additional degrees of freedom d o f the dynamic explicit global resolution scheme as well as the local integration scheme to compute the stress tensor and the state variables at each integration point of each element are presented application is made to the typical uniaxial tension specimen under plane strain conditions in order to chow the predictive capabilities of the proposed micromorphic model particularly against its ability to give at convergence a mesh independent solution even for high values of the ductile damage i e the macroscopic cracks

free monotone transport without a trace we adapt the free monotone transport results of guionnet and shlyakhtenko to the type iii case as a direct application we obtain that the q deformed araki woods algebras are isomorphic for sufficiently small q

a regularized phenomenological multiscale damage model summary we present a regularized phenomenological multiscale model where elastic properties are computed using direct homogenization and subsequently evolved using a simple three parameter orthotropic continuum damage model the salient feature of the model is a unified regularization framework based on the concept of effective softening strain the unified regularization scheme has been employed in the context of constitutive law rescaling and the staggered nonlocal approach we show that an element erosion technique for crack propagation when exercised with one of the two regularization schemes 1 possesses a characteristic length 2 is consistent with fracture mechanics approach and 3 does not suffer from pathological mesh sensitivity copyright 2014 john wiley sons ltd

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

dislocation microstructures and the effective behavior of single crystals we consider single crystal plasticity in the limiting case of infinite latent hardening which signifies that the crystal must deform in single slip at all material points this requirement introduces a nonconvex constraint and thereby induces the formation of fine scale structures we restrict attention throughout to linearized kinematics and deformation theory of plasticity which is appropriate for monotonic proportional loading and confers the boundary value problem of plasticity a well defined variational structure analogous to elasticity we first study a scale invariant local problem we show that by developing microstructures in the form of sequential laminates of finite depth crystals can beat the single slip constraint i e the macroscopic relaxed constitutive behavior is indistinguishable from multislip ideal plasticity in a second step we include dislocation line energies and hence a length scale into the model different regimes lead to several possible types of microstructure patterns we present constructions which achieve the various optimal scaling laws and discuss the relation with experimentally known scalings such as the hall petch law

variational gradient plasticity at finite strains part i mixed potentials for the evolution and update problems of gradient extended dissipative solids this work outlines a theoretical and computational framework of finite inelasticity with length scales based on a rigorous exploitation of mixed variational principles in contrast to classical local approaches to inelasticity based on locally evolving internal variables order parameter fields are taken into account governed by additional balance type partial differential equations including micro structural boundary conditions this incorporates non local dissipative effects based on length scales which reflect properties of the material micro structure typical examples are phase field theories gradient damage and strain gradient plasticity we outline unified minimization and saddle point principles for the evolution problem of first order gradient type standard dissipative solids particular emphasis is put on mixed multi field representations where both the microstructural variable itself as well as its dual driving force are present these settings are needed for models with threshold functions formulated in the space of the driving forces in particular for finite gradient plasticity the central aim is to define constitutive rate type and algorithmic incremental potentials whose variational derivatives govern the coupled macro and micro balances in both the continuous as well as the time discrete setting their existence underlines the inherent symmetry of standard dissipative solids we demonstrate geometrically consistent constructions of these potentials for important classes of finite inelasticity providing a fresh look on models of multiplicative and additive gradient plasticity of single crystals and amorphous materials the potentials provide maximum compact representations of those complex material models are the cornerstones of a subsequent mixed finite element design and should be considered as the primary objects for the theoretical and computational modeling

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

nonlinear regularization operators as derived from the micromorphic approach to gradient elasticity viscoplasticity and damage the construction of regularization operators presented in this work is based on the introduction of strain or damage micromorphic degrees of freedom in addition to the displacement vector and of their gradients into the helmholtz free energy function of the constitutive material model the combination of a new balance equation for generalized stresses and of the micromorphic constitutive equations generates the regularization operator within the small strain framework the choice of a quadratic potential w r t the gradient term provides the widely used helmholtz operator whose regularization properties are well known smoothing of discontinuities at interfaces and boundary layers in hardening materials and finite width localization bands in softening materials the objective is to review and propose nonlinear extensions of micromorphic and strain damage gradient models along two lines the first one introducing nonlinear relations between generalized stresses and strains the second one envisaging several classes of finite deformation model formulations the generic approach is applicable to a large class of elastoviscoplastic and damage models including anisothermal and multiphysics coupling two standard procedures of extension of classical constitutive laws to large strains are combined with the micromorphic approach additive split of some lagrangian strain measure or choice of a local objective rotating frame three distinct operators are finally derived using the multiplicative decomposition of the deformation gradient a new feature is that a free energy function depending solely on variables defined in the intermediate isoclinic configuration leads to the existence of additional kinematic hardening induced by the gradient of a scalar micromorphic variable

slow motion and metastability for a nonlocal evolution equation in this paper we consider a nonlocal evolution equation in one dimension which describes the dynamics of a ferromagnetic system in the mean field approximation in the presence of a small magnetic field it admits two stationary and homogeneous solutions representing the stable and metastable phases of the physical system we prove the existence of an invariant one dimensional manifold connecting the stable and metastable phases this is the unstable manifold of a distinguished spatially nonhomogeneous stationary solution called the critical droplet 4 10 we show that the points on the manifold are droplets longer or shorter than the critical one and that their motion is very slow in agreement with the theory of metastable patterns we also obtain a new proof of the existence of the critical droplet which is supplied with a local uniqueness result

a semi finite algebra associated to a subfactor planar algebra we canonically associate to any planar algebra two type ii factors m the subfactors constructed previously by the authors in guionnet et al 2010 6 are isomorphic to compressions of m to finite projections we show that each m is isomorphic to an amalgamated free product of type i von neumann algebras with amalgamation over a fixed discrete type i von neumann subalgebra in the finite depth case existing results in the literature imply that m m is the amplification a free group factor on a finite number of generators as an application we show that the factors mj constructed in guionnet et al in press 6 are isomorphic to interpolated free group factors l f rj rj 1 2 2j 1 i where 2 is the index of the planar algebra and i is its global index other applications include computations of laws of jones wenzl projections

free monotone transport by solving a free analog of the monge ampere equation we prove a non commutative analog of brenier s monotone transport theorem if an n tuple of self adjoint non commutative random variables z 1 z n satisfies a regularity condition its conjugate variables 1 n should be analytic in z 1 z n and j should be close to z j in a certain analytic norm then there exist invertible non commutative functions f j of an n tuple of semicircular variables s 1 s n so that z j f j s 1 s n moreover f j can be chosen to be monotone in the sense that and g is a non commutative function with a positive definite hessian in particular we can deduce that c z 1 z n c s 1 s n and w z 1 dots z n cong l mathbb f n thus our condition is a useful way to recognize when an n tuple of operators generate a free group factor we obtain as a consequence that the q deformed free group factors vargamma q mathbb r n are isomorphic for sufficiently small q with bound depending on n to free group factors we also partially prove a conjecture of voiculescu by showing that free gibbs states which are small perturbations of a semicircle law generate free group factors lastly we show that entrywise monotone transport maps for certain gibbs measure on matrices are well approximated by the matricial transport maps given by free monotone transport

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

a new approach for a nonlocal nonlinear conservation law we describe an approach to nonlocal nonlinear advection in one dimension that extends the usual pointwise concepts to account for nonlocal contributions to the flux the spatially nonlocal operators we consider do not involve derivatives instead the spatial operator involves an integral that in a distributional sense reduces to a conventional nonlinear advective operator in particular we examine a nonlocal inviscid burgers equation which gives a basic form with which to characterize properties associated with well posedness and to examine numerical results for specific cases we describe the connection to a nonlocal viscous regularization which mimics the viscous burgers equation in an appropriate limit we present numerical results that compare the behavior of the nonlocal burgers formulation to the standard local case the developments presented in this paper form the preliminary building blocks upon which to build a theory of nonlocal advection phenomena consistent within the peridynamic theo

free transport for interpolated free group factors abstract in this article we study a form of free transport for the interpolated free group factors extending the work of guionnet and shlyakhtenko for the usual free group factors 11 our model for the interpolated free group factors comes from a canonical finite von neumann algebra m associated to a finite connected weighted graph v e 12 13 with this model we use an operator valued version of voiculescu s free difference quotient introduced in 13 to state a schwinger dyson equation which is valid for the generators of m we construct free transport for appropriate perturbations of this equation also m can be constructed using the machinery of shlyakhtenko s operator valued semicircular systems 24

an implicit tensorial gradient plasticity model formulation and comparison with a scalar gradient model many rate independent models for metals utilize the gradient of effective plastic strain to capture size dependent behavior this enhancement sometimes termed as explicit gradient formulation requires higher order tractions to be imposed on the evolving elasto plastic boundary and the resulting numerical framework is complicated an implicit scalar gradient model was thus developed in peerlings peerlings r h j 2007 on the role of moving elastic plastic boundaries in strain gradient plasticity model simul mater sci eng 15 109 120 that has only c0 continuity requirements and its implementation is straightforward however both explicit and implicit scalar gradient models can be problematic when the effective plastic strains do not have smooth profiles to address this limitation an implicit tensorial gradient model is proposed in this paper based on the generalized micromorphic framework it is also demonstrated that the scalar and tensorial implicit gradient models give similar results when the effective plastic strains fluctuate smoothly

nonlinear microstrain theories a hierarchy of higher order continua is presented that introduces additional degrees of freedom accounting for volume changes rotation and straining of an underlying microstructure an increase in the number of degrees of freedom represents a refinement of the material description in addition to available nonlinear cosserat and micromorphic theories general formulations of elastoviscoplastic behaviour are proposed for microdilatation and microstretch continua a microstrain theory is introduced that is based on six additional degrees of freedom describing the pure straining of the microstructural element in each case balance equations and boundary conditions are derived decompositions of the finite strain measures into elastic and plastic parts are provided the formulation of finite deformation elastoviscoplastic constitutive equations relies on the introduction of the free energy and dissipation potentials thus complying with requirements of continuum thermodynamics some guidelines for the selection of a suitable higher order model for a given material close the discussion

on a theory of nonlocal elasticity of bi helmholtz type and some applications abstract a theory of nonlocal elasticity of bi helmholtz type is studied we employ eringen s model of nonlocal elasticity with bi helmholtz type kernels to study dispersion relations screw and edge dislocations the nonlocal kernels are derived analytically as green functions of partial differential equations of fourth order this continuum model of nonlocal elasticity involves two material length scales which may be derived from atomistics the new nonlocal kernels are nonsingular in one two and three dimensions furthermore the nonlocal elasticity of bi helmholtz type improves the one of helmholtz type by predicting a dispersion relationship with zero group velocity at the end of the first brillouin zone new solutions for the stresses and strain energy of screw and edge dislocations are found

large strain finite element analysis of a local second gradient model application to localization summary a large strainnite element formulation based on a local second gradient plasticity model is presented the corresponding constitutive equations were developed as a direct extension of microstructured siam j appl math 1973 25 3 556 575 arch rational mech anal 1964 16 51 78 or micromorphic in mechanics of generalized continua iutam symposium kroner ed springer berlin 1968 18 35 j math mech 1966 15 6 909 924 continua in which a mathematical constraint between the micro kinematics description and the usual macrodeformation gradienteld has been introduced this constraint is enforced in a weak sense by the use of lagrange multipliers in order to avoid diculties with the c 1 continuity for thenite element method correspondingnite elements are then constructed involving the lagrange multiplierseld a geometrically non linear 2 dnite element code is developed within a framework of an incremental method for every step a full newton raphson involving a numerical consistent tangent stiness operator for the complete model i e the second gradient terms as well as the classical ones is done and some numerical tests allow to validate the method and to discuss the inuence of the geometrical non linearity copyright 2002 john wiley sons ltd

gradient damage models toward full scale computations abstract the description of localisation and structural failure through local damage constitutive laws leads to ill posed problems to retrieve a well posed problem the material state is described through the gradients of damage in addition to the strain and the damage under some assumptions it is shown that these gradient laws can be cast into a variational formulation its numerical treatment is reduced to the design of a mixed finite element so that algorithms such as newton s method and path following techniques are still available possible convergence difficulties related to the reduction of the damage band width and the appearance of micro instabilities are removed thanks to the design of a new class of brittle laws finally numerical applications show the convergence with mesh refinement the compatibility with coarser approaches such as griffith s theory and cohesive crack models the robustness in case of unstable multi cracking the applicability to full scale 3d situations and the physical relevance compared to experimental data

numerical simulation of ductile fracture with the rousselier constitutive law this contribution aims at developing robust and reliable tools dedicated to predictive numerical simulation of ductile fracture description of damage mechanisms relies here on the rousselier constitutive law to achieve robustness and objectivity several points need then to be addressed reduction of the set of internal variables so as to recast the model into the framework of generalised standard materials proper expression of the normal plastic flow rule at the vertex of the yield surface finite strain formulation based on a multiplicative split of the deformation gradient fully implicit integration of the constitutive equations leading to a single scalar equation that admits a unique root use of mixed finite elements to avoid volumetric locking strain gradient based nonlocal model to control strain localisation and remedy spurious mesh dependency the capabilities of the resulting numerical formulation are demonstrated through simulating the cup cone fracture of a notched tensile specimen

anisotropic additive plasticity in the logarithmic strain space modular kinematic formulation and implementation based on incremental minimization principles for standard materials the paper presents a modular formulation and computational implementation of a class of anisotropic plasticity models at finite strains based on incremental minimization principles the modular kinematic setting consists of a constitutive model in the logarithmic strain space that is framed by a purely geometric pre and postprocessing on the theoretical side the point of departure is an a priori six dimensional approach to finite plasticity based on the notion of a plastic metric in a first step a geometric preprocessor defines a total and a plastic logarithmic strain measure obtained from the current and the plastic metrics respectively in a second step these strains enter in an additive format a constitutive model of anisotropic plasticity that may have a structure identical to the geometrically linear theory the model defines the stresses and consistent tangents work conjugate to the logarithmic strain measure in a third step these objects of the logarithmic space are then mapped back to nominal lagrangian or eulerian objects by a geometric postprocessor this geometric three step approach defines a broad class of anisotropic models of finite plasticity directly related to counterparts of the geometrically linear theory it is specified to a model problem of anisotropic metal plasticity on the computational side we develop an incremental variational formulation of the above outlined constitutive structure where a quasi hyperelastic stress potential is obtained from a local constitutive minimization problem with respect to the internal variables it is shown that this minimization problem is exclusively restricted to the logarithmic strain space in a structure identical to the small strain theory the minimization problem determines the internal state of the material for finite increments of time we develop a discrete formulation in terms of just one scalar parameter for the amount of incremental flow the existence of the incremental stress potential provides a natural basis for the definition of the geometric postprocessor based on function evaluations furthermore the global initial boundary value problem of the elastic plastic solid appears in the incremental setting as an energy minimization problem numerical examples show that the results obtained are surprisingly close to those obtained by a reference framework of multiplicative plasticity