Extended Finite Element Method - دانشگاه تهران

BLUK134-FMP1: PBUPrinter: Yet to ComeAugust 21, 200718:35MohammadiPreface‘I am always obliged to a person who has taught me a single word.’Progressive failure/fracture analysis of structures has been an active research topic forthe past two decades. Historically, it has been addressed either within the frameworkof continuum computational plasticity and damage mechanics, or the discontinuousapproach of fracture mechanics. The present form of linear elastic fracture mechanics(LEFM), with its roots a century old has since been successfully applied to variousclassical crack and defect problems. Nevertheless, it remains relatively limited to simplegeometries and loading conditions, unless coupled with a powerful numerical tool suchas the finite element method and meshless approaches.The finite element method (FEM) has undoubtedly become the most popular andpowerful analytical tool for studying a wide range of engineering and physical problems. Several general purpose finite element codes are now available and concepts ofFEM are usually offered by all engineering departments in the form of postgraduateand even undergraduate courses. Singular elements, adaptive finite element procedures,and combined finite/discrete element methodologies have substantially contributed tothe development and accuracy of fracture analysis of structures. Despite all achievements, the continuum basis of FEM remained a source of relative disadvantage fordiscontinuous fracture mechanics. After a few decades, a major breakthrough seemsto have been made by the fundamental idea of partition of unity and in the form of theeXtended Finite Element Method (XFEM).This book has been prepared primarily to introduce the concepts of the newlydeveloped extended finite element method for fracture analysis of structures. An attempt has also been made to discuss the essential features of XFEM for other relatedengineering applications. The book can be divided into four parts. The first part is dedicated to the basic concepts and fundamental formulations of fracture mechanics. Itcovers discussions on classical problems of LEFM and their extension to elastoplasticfracture mechanics (EPFM). Issues related to the standard finite element modellingof fracture mechanics and the basics of popular singular finite elements are reviewedbriefly.The second part, which constitutes most of the book, is devoted to a detailed discussion on various aspects of XFEM. It begins by discussing fundamentals of partitionof unity and basics of XFEM formulation in Chapter 3. Effects of various enrichmentfunctions, such as crack tip, Heaviside and weak discontinuity enrichment functions arealso investigated. Two commonly used level set and fast marching methods for tracking moving boundaries are explained before the chapter is concluded by examining anumber of classical problems of fracture mechanics. The next chapter deals with theorthotropic fracture mechanics as an extension of XFEM for ever growing applicationsix

BLUK134-FMxP1: PBUPrinter: Yet to ComeAugust 21, 200718:35MohammadiPrefaceof composite materials. A different set of enrichment functions for orthotropic mediais presented, followed by a number of simulations of benchmark orthotropic problems.Chapter 5, devoted to simulation of cohesive cracks by XFEM, provides theoreticalbases for cohesive crack models in fracture mechanics, classical FEM and XFEM.The snap-back response and the concept of critical crack path are studied by solving anumber of classical cohesive crack problems.The third part of the book (Chapter 6) provides basic information on new frontiersof application of XFEM. It begins with discussions on interface cracking, which includeclassical solutions from fracture mechanics and XFEM approximation. Application ofXFEM for solving contact problems is explained and numerical issues are addressed.The important subject of dynamic fracture is then discussed by introducing classicalformulations of fracture mechanics and the recently developed idea of time–spacediscretization by XFEM. New extensions of XFEM for very complex applications ofmultiscale and multiphase problems are explained briefly.The final chapter explains a number of simple instructions, step-by-step procedures and algorithms for implementing an efficient XFEM. These simple guidelines, incombination with freely available XFEM source codes, can be used to further advancethe existing XFEM capabilities.This book is the result of an infinite number of brilliant research works in thefield of computational mechanics for many years all over the world. I have tried toappropriately acknowledge the achievements of corresponding authors within the text,relevant figures, tables and formulae. I am much indebted to their outstanding researchworks and any unintentional shortcoming in sufficiently acknowledging them is sincerely regretted. Perhaps such a title should have become available earlier by one ofthe pioneers of the method, i.e. Professor T. Belytschko, a shining star in the universeof computational mechanics, Dr J. Dolbow, Dr N. Moës, Dr N. Sukumar and possiblyothers who introduced, contributed and developed most of the techniques.I would like to extend my acknowledgement to Blackwell Publishing Limited,for facilitating the publication of the first book on XFEM; in particular N. WarnockSmith, J. Burden, L. Alexander, A. Cohen and A. Hallam for helping me throughoutthe work. Also, I would like to express my sincere gratitude to my long-time friend,Professor A.R. Khoei, with whom I have had many discussions on various subjects ofcomputational mechanics, including XFEM. Also my special thanks go to my students:Mr A. Asadpoure, to whom I owe most of Chapter 4, Mr S.H. Ebrahimi for solvingisotropic examples in Chapter 3 and Mr A. Forghani for providing some of the resultsin Chapter 5.This book has been completed on the eve of the new Persian year; a ‘temporalinterface’ between winter and spring, and an indication of the beginning of a bloomingseason for XFEM, I hope.Finally, I would like to express my gratitude to my family for their love, understanding and never-ending support. I have spent many hours on writing this book; hoursthat could have been devoted to my wife and little Sogol: the spring flowers that inspirethe life.Soheil MohammadiTehran, IranSpring 2007

BLUK134-FMP1: PBUPrinter: Yet to ComeAugust 21, 200718:35MohammadiNomenclatureααcα f , αsβγspγse , γsγx yδδδ(ξ )δi jεε f , εcεijε̄ijεijauxεvεyldηθθκ, κ λλμν, νijξξ(x)ρρρ f , ρcρintσσ f , σcσgtipσtσ0Curvilinear coordinateLoad factor for cohesionThermal diffusivity of fluid and solid phasesCurvilinear coordinateSurface energy densityElastic and plastic surface energiesEngineering shear strainPlastic crack tip zoneVariation of a functionDirac delta functionKronecker delta functionStrain tensorStrain field at fine and coarse scalesStrain componentsDimensionless angular geometric functionAuxiliary strain componentsKinetic mobility coefficientYield strainLocal curvilinear (mapping) coordinate systemCrack propagation angle with respect to initial crackAngular polar coordinateMaterial parametersLame modulusEigenvalue of the characteristic equationShear modulusIsotropic and orthotropic Poisson’s ratiosLocal curvilinear (mapping) coordinate systemDistance functionRadius of curvatureDensityDensity of fine and coarse scalesCurvature of the propagating interfaceStress tensorStress field at fine and coarse scalesStress tensor at a Gauss pointNormal tensile stress at crack tipApplied normal tractionxi

BLUK134-FMP1: PBUPrinter: Yet to ComeAugust 21, jauxσnσnσyldττ0τcτnτnφ(x)φ(z)φs (z)ϕϕχ (x)χ (z)ψ (x)ψ (z)ω c t u Ξ j (x) (x) f , c f , s puCritical stress for crackingStress componentsDimensionless angular geometric functionAuxiliary stress componentsStress component normal to an interfaceStress component at time step nYield stressDeviatoric stressApplied tangential tractionCohesive shear tractionTime functionsDeviatoric stress tensor at time step nLevel set functionComplex stress functionStress function for shear problemAngle of orthotropic axesPhase angle for interface fractureEnrichment function for weak discontinuitiesStress functionEnrichment functionStress functionOscillation indexBoundaryCrack boundaryTraction (natural) boundaryDisplacement (essential) boundaryFinite variation of a functionCoefficient matrixHomogenisation/average operatorPotential energyMoving least squares shape functionsStress functionDomainFine and course scale domainsFluid and solid domainsDomain associated with the partition of unityaaab, afahaiakA bbbiCrack length/half lengthSemi-major axis of ellipseBackward and forward indexes in fast marching methodHeaviside enrichment degrees of freedomEnrichment degrees of freedomEnrichment degrees of freedomArea associated with the domain J integralWidth of a plateSemi-minor axis of ellipseCrack tip enrichment degrees of freedom

BLUK134-FMP1: PBUPrinter: Yet to ComeAugust 21, BibcccijcRc f , csCdd/dtDDc , D fDlocD/DtDxb , DxfE, E iE ftf (r )ff riMatrix of derivatives of shape functionsMatrix of derivatives of final shape functionsB matrix for coarse scaleB matrix for fine scaleStrain–displacement matrix (derivatives of shape functions)Strain–displacement matrix (derivatives of shape functions)Matrix of derivatives of enrichment (Heaviside) of shape functionsMatrix of derivatives of enrichment (crack tip) of shape functionsConstant parameterSize of crack tip contour for J integralMaterial constantsRayleigh speedSpecific heat for fluid and solid phasesMaterial constitutive matrixDistanceTime derivativeMaterial modulus matrixMaterial modulus in coarse and fine scalesLocalisation modulusMaterial time derivativeBackward and forward finite difference approximationsIsotropic and orthotropic Young’s modulusMaterial parameterUniaxial tensile strengthRadial functionNodal force vectorNodal force components (classic and enriched)fbftfcf cohf extf uintf aintFli (x)gg(θ)g j (θ )GGG1, G2dynGIH (ξ)HliJJBody force vectorExternal traction vectorCohesive crack traction vectorCohesive nodal force vectorExternal force vectorInternal nodal force vector due to external loadingInternal nodal force vector due to cohesive forceCrack tip enrichment functionsApplied gravitational body forceAngular function for a crack tip kink problemOrthotropic crack tip enrichment functionsShear modulusFracture energy release rateMode I and II fracture energy release ratesDynamic mode I fracture energy releaseHeaviside functionLatent heatComplex number, i 2 1Jacobian matrixJ integral

BLUK134-FMP1: PBUPrinter: Yet to ComeAugust 21, 200718:35MohammadiNomenclaturexivJ actJ auxJkk0k0 , k1 , k2 , k3 , k4kiks , k fkn , ktKKhomKrijsKKCK eqK I , K II , K IIIK̄ I , K̄ IIK Iaux , K IIauxK Ic , K IIcK IcohesionK N̄ jp(x)pp̄PiqqActual J integralAuxiliary J integralMode k contour integral JDimensionless constant for the power hardening lawConstant coefficientsConductivity coefficient for phase iThermal conductivity for solid and fluid phasesNormal/tangential interface propertiesStiffness matrixHomogenised stiffness matrixStiffness matrix componentsStress intensity factorCritical stress intensity factorEquivalent mixed mode stress intensity factorMode I , II and III stress intensity factorsNormalized mode I and mode II stress intensity factorsAuxiliary mode I and mode II stress intensity factorsCritical mode I and mode II stress intensity factorsCohesive mode I stress intensity factorCrack mode I stress intensity factorDynamic mode I stress intensity factorCharacteristic lengthCharacteristic length for crack propagaionNumber of enrichment functionsNumber of nodes to be enriched by crack tip enrichment functionsNumber of crack tip enrichment functionsMach numberInteraction integralTotal massLumped mass componentMass matrix componentPower number for the plastic model (Section 2.6.4)Number of Gauss pointsNumber of nodes within each moving least squares support domainNumber of independent domains of partition of unityNumber of nodes in a finite elementNormal vectorNormal vector to an internal interfaceMatrix of shape functionsShape functionNew set of generalised finite element method shape functionsBasis functionHydrostatic pressurePredefined hydrostatic pressureLoading condition iArbitrary smoothing functionHeat flux

BLUK134-FMP1: PBUPrinter: Yet to ComeAugust 21, 200718:35MohammadiNomenclatureqiQQijrrgrpRs f , ssSaScSdtttintTT f , TsTi (t)Tmuûu̇ūu̇ üuiauxucohuenruFEuh (x)ū jũ jupux , u yU 1, U 2UkUspUse , UsU vv̄vnv1 , v2V(t)wwcWW auxNodal values of the arbitrary smoothing functionInput heat to systemMatrix of homogenous anisotropic solidsRadial distance/coordinateRadial distance of a Gauss point from crack tipCrack tip plastic zoneRamp functionHeat source for fluid and solid phasesSet of accepted nodesSet of candidate nodesSet of distant nodesTimeTractionSurface traction along internal boundaryTemperatureTemperature of fluid and solid phasesTime shape functionsMelting/fusion temperatureDisplacement vectorLocal symmetric displacement vectorVelocity vectorPrescribed displacementPrescribed velocityAcceleration vectorAuxiliary displacement fieldDisplacement field obtained from crack surface tractionsEnriched displacement fieldClassical finite element displacement fieldApproximated displacement fieldNodal displacement vectorTransformed displacementPeriodic displacementxand y displacement componentSymmetric and antisymmetric crack tip displacementsKinetic energyStrain energyElastic and plastic strain energiesSurface energyVelocity vectorPrescribed velocityNormal interface speedLongitudinal and shear wave velocitiesVector of approximated velocity degrees of freedomCrack openingCritical crack openingExternal workAuxiliary workxv

BLUK134-FMP1: PBUPrinter: Yet to ComeAugust 21, 200718:35MohammadiNomenclaturexviW cohW extWgWgrW intWMWsWt (t)xxc , x fx x1 , x2x y z x iyz̄ x i yziVirtual work of cohesive forceVirtual work of external loadingGauss weight factorRadial weight function at a Gauss point gInternal virtual workInteraction workStrain energyTime weight functionPosition vectorPosition vector for coarse and fine scalesPosition of projection point on an interfaceTwo-dimensional coordinate systemLocal crack tip coordinate axesLocal crack tip coordinate axesComplex variableConjugate complex variableComplex parametersf , f f , f x First and second derivative of a functionFirst and second integrals of a functionNabla operatorJump operator across an MMFPMFPZGFEMHRRLEFMLSMMCCMEPUMLPGMLSBoundary Element MethodCharacteristic Based SplitCrack Opening DisplacementCrack Tip Opening DisplacementDisplacement Correlation TechniqueDegree Of FreedomEquivalent Domain IntegralElement-Free GalerkinEquilibrium On LineElastic Plastic Fracture MechanicsFinite Difference MethodFinite ElementFinite Element MethodFast Marching MethodFinite Point MethodFracture Process ZoneGeneralised Finite Element MethodHutchinson–Rice–RosengrenLinear Elastic Fracture MechanicsLevel Set MethodModified Crack ClosureMultiscale Enrichment Partition of UnityMeshless Local Petrov–GalerkinMoving Least Squares

BLUK134-FMP1: PBUPrinter: Yet to ComeAugust 21, ARSIFSPHTFEMTXFEMWLSXFEM, X-FEMNon-Uniform Rational B-SplineOrdered Upwind MethodPartition of UnityPartition of Unity Finite Element MethodReproducing Kernel Particle MethodStatically Admissible stress RecoveryStress Intensity FactorSmoothed Particle HydrodynamicsTime Finite Element MethodTime eXtended Finite Element MethodWeighted Least SquareseXtended Finite Element Methodxvii

Chapter 1Introduction1.1 ANALYSIS OF STRUCTURESThe finite element method (FEM) has undoubtedly become the most popular andpowerful analytical tool for studying the behaviour of a wide range of engineering andphysical problems. Several general purpose finite element softwares have been developed,verified and calibrated over the years and are now available to almost anyone who asksand pays for them. Furthermore, concepts of FEM are usually offered by all engineeringdepartments in the form of postgraduate and even undergraduate courses.One of the important applications of FEM is the analysis of crack propagationproblems. Fundamentals of the present form of the linear elastic fracture mechanics(LEFM) came to the existence practically in naval laboratories during the First WorldWar. Since then, LEFM has been successfully applied to various classical crack anddefect problems, but remained relatively limited to simple geometries and loadingconditions.Introduction and fast development of the finite element method drastically changedthe extent of application of LEFM. FEM virtually had no limitation in solving complexgeometries and loading conditions, and soon it was extended to nonlinear materials andlarge deformation problems (Zienkiewicz et al. 2005). As a result, LEFM could now relyon a powerful analytical tool in order to determine its fundamental concepts andgoverning criteria such as the crack energy release rate and the stress intensity factor forany complex problem. General LEFM stability criteria could then be used to assess thestability/propagation of an existing crack.Application of FEM into linear elastic fracture mechanics and its extension to elasticplastic fracture mechanics (EPFM) has now expanded to almost all crack problems.Parametric studies and experimental observations have even resulted in the introductionof new design codes for containing a stable crack. However, the essence of analysesremained almost unchanged: LEFM basic concepts combined with classical continuumbased FEM techniques through smeared or discrete crack models.After a few decades, a major breakthrough seemed to be evolving in the fundamentalidea of partition of unity and in the form of the eXtended Finite Element Method (X-FEMor XFEM).1Trim added in PDF - Aptara

2 Extended Finite Element Method1.2 ANALYSIS OF DISCONTINUITIESProgressive failure/fracture analysis of structures has been an active subject of researchfor many years. Historically, it was addressed either within the framework of continuummechanics, including computational plasticity and damage mechanics, or thediscontinuous approach of fracture mechanics (Owen and Hinton 1980).These methods, however, are applied to fundamentally different classes of failureproblems. While the theory of plasticity and damage mechanics are basically designed forproblems where the displacement field and usually the strain field remain continuouseverywhere (continuous problems), fracture mechanics is essentially formulated to dealwith strong discontinuities (cracks) where both the displacement and strain fields arediscontinuous across a crack surface (Fig. 1.1) (Mohammadi 2003).Figure 1.1 Different categories of continuities.In practice, fracture mechanics is also used for weak discontinuity problems, and bothdamage mechanics and the theory of plasticity have been modified and adapted forfailure/fracture analysis of structures with strong discontinuities. It is, therefore, difficultto distinguish between the practical engineering applications exclusively associated witheach class of analytical methods.Inclusion of some basic concepts from fracture mechanics such as non-local models,energy release rate, softening models in combination with adaptive remeshing techniqueshave allowed for successful simulations of crack problems with a certain level ofaccuracy.Trim added in PDF - Aptara

Introduction 31.3 FRACTURE MECHANICSFundamental concepts of fracture mechanics can be traced back to the late nineteenth andearly twentieth centuries. Both experimental observations and theoretical elasticity helpedto create the fundamental aspects of

3.5.3 Partition of unity finite element method 72 3.5.4 Generalised finite element method 73 3.5.5 Extended finite element method 73 3.5.6 Hp-clouds enrichment 73 3.5.7 Generalisation of the PU enrichment 74 3.5.8 Transition from standard to enriched approximation 74 3.6 ISOTROPIC XFEM 76 3.6.1 Basic XFEM approximation 76