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أخوانى فى الله أحضرت لكم كتاب Non-linear Finite Element Analysis of Solids and Structures Volume 2: Advanced Topics M.A. Crisfield Imperial College of Science, Technology and Medicine, London, UK
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Contents Preface xiii 10 More continuum mechanics 1 10.1 Relationships betweensome strain measures and the structures 10.2 Large strains and the Jaumann rate 10.3 Hyperelasticity 10.4 The Truesdell rate 10.5 Conjugate stress and strain measures with emphasison isotropic conditions 10 10.6 Furtherwork on conjugate stress and strain measures 13 10.6.1 Relationship betweeni: and U 14 10.6.2 Relationship between the Bio! stress, B and the Kirchhoff stress, T 15 10.6.3 Relationship betweenU, the i’s and the spin of the Lagrangian triad, W, 15 10.6.4 Relationship between€, the A’s and the spin, W, 16 10.6.5 Relationship between6,the 2’s andthe spin, W, 17 10.6.6 Relationship between€and E 17 10.6.6.1 Specific strain measures 17 10.6.7 Conjugate stress measures 18 10.7 Using log,V with isotropy 19 10.8 Other stress rates and objectivity 20 10.9 Special notation 22 10.10 References 24 11 Non-orthogonal coordinates and CO- and contravariant tensor components 26 1 1.1 Non-orthogonal coordinates 26 11.2 Transforming the componentsof a vector (first-ordertensor) to a new set of base vectors 28 11.3 Second-order tensors in non-orthogonal coordinates 30 11.4 Transformingthe componentsof a second-order tensor to a new set of base vectors 30 11.5 The metrictensor 31 11.6 Work terms and the trace operation 32vi CONTENTS 11.7 Covariant components, natural coordinates and the Jacobian 33 11.8 Green’s strain and the deformation gradient 35 11.8.1 Recoveringthe standard cartesian expressions 35 11.9 The second Piola-Kirchhoff stresses and the variation of the Green’s strain 36 11.10 Transforming the components of the constitutive tensor 37 11.11 A simple two-dimensional example involving skew coordinates 38 11.12 Special notation 42 11.13 References 44 12 More finite element analysisof continua 45 12.1 A summary of the key equations for the total Lagrangian formulation 46 12.1.1 The internalforce vector 46 12.1.2 The tangentstiffnessmatrix 47 12.2 The internal force vector for the ‘Eulerian formulation’ 47 12.3 The tangent stiffness matrix in relation to the Truesdell rate of Kirchhoff stress 49 12.3.1 Continuum derivationof the tangentstiffnessmatrix 49 12.3.2 Discretised derivationof the tangent stiffnessmatrix 51 12.4 The tangent stiffness matrix using the Jaumann rate of Kirchhoff stress 53 12.4.1 Alternativederivationof the tangent stiffnessmatrix 54 12.5 The tangent stiffness matrix using the Jaumann rate of Cauchy stress 55 12.5.1 Alternative derivationof the tangentstiffnessmatrix 56 12.6 Convected coordinates and the total Lagrangian formulation 57 12.6.1 Elementformulation 57 12.6.2 The tangentstiffnessmatrix 59 12.6.3 Extensionsto three dimensions 59 12.7 Special notation 60 12.8 References 61 13 Large strains, hyperelasticityand rubber 62 13.1 Introduction to hyperelasticity 62 13.2 Using the principal stretch ratios 63 13.3 Splitting the volumetric and deviatoric terms 65 1 3.4 Development using second Piola-Kirchhoff stresses and Green’s strains 66 13.4.1 Plane strain 69 13.4.2 Plane stress with incompressibility 69 13.5 Total Lagrangian finite element formulation 71 13.5.1 A mixed formulation 72 12.5.2 A hybridformulation 74 13.6 Developments using the Kirchhoff stress 76 13.7 A ‘Eulerian’ finite element formulation 78 13.8 Working directly with the principal stretch ratios 79 13.8.1 The compressible ‘neo-Hookean model’ 80 13.8.2 Usingthe Green strain relationships in the principal directions 81 13.8.3 Transforming the tangentconstitutive relationshipsfor a ‘Eulerian formulation’ 84 13.9 Examples 86 13.9.1 A simpleexample 86 13.9.2 The compressible neo-Hookeanmodel 89 13.10 Further work with principal stretch ratios 89 13.10.1 An enerav function usina the DrinciPalloa strains fthe Henckvmodel) 90CONTENTS vii 13.10.2 Ogden’s energy function 91 13.10.3 An example using Hencky’s model 93 13.11 Special notation 95 13.12 References 97 14 More plasticityand other material non-linearity-I 99 14.1 Introduction 99 14.2 Other isotropicyield criteria 99 14.2.1 The flow rules 104 14.2.2 The matrix ?a/(% 105 14.3 Yield functions with corners 107 14.3.1 A backward-Euler return with two activeyield surfaces 107 14.3.2 A consistent tangent modularmatrix with two active yield surfaces 108 14.4 Yield functions for shells that use stress resultants 109 14.4.1 The one-dimensional case 109 14.4.2 The two-dimensional case 112 14.4.3 A backward-Euler return with the lllyushin yieldfunction 113 14.4.4 A backward-Euler return and consistent tangent matrix for the llyushinyield criterionwhen two yield surfaces are active 114 14.5 Implementing a form of backward-Euler procedurefor the Mohr-Coulomb yield criterion 115 14.5.1 Implementing a two-vectored return 118 14.5.2 A return from a corner or to the apex 119 14.5.3 A consistent tangent modular matrix following a single-vector return 120 14.5.4 A consistent tangent matrix following a two-vectored return 121 14.5.5 A consistent tangent modular matrix following a return from a corner or an apex 121 14.6 Yield criteriafor anisotropic plasticity 122 14.6.1 Hill’s yield criterion 122 14.6.2 Hardening with Hill’syieldcriterion 124 14.6.3 Hill’s yield criterion for plane stress 126 14.7 Possible return algorithms and consistent tangent modular matrices 129 14.7.1 The consistent tangent modularmatrix 130 14.8 Hoffman’s yield criterion 131 14.8.1 The consistent tangent modularmatrix 133 14.9 The Drucker-Prager yieldcriterion 133 14.10Usingan eigenvector expansionfor the stresses 134 14.10.1 An example involving plane-stress plasticity and the von Mises yieldcriterion 135 14.11 Cracking, fracturing and softening materials 135 14.11.1 Mesh dependency and alternative equilibrium states 135 14.11.2 ‘Fixed’ and ‘rotating’ crack modelsin concrete 140 14.11.3 Relationship between the ‘rotating crack model’ and a ‘deformation theory’ plasticity approach usingthe ‘squareyield criterion’ 142 14.11.4 A flow theory approachfor the ‘square yield criterion’ 144 14.12 Damage mechanics 148 14.13 Special notation 152 14.14 References 154 15 More plasticityand other materialnon-linearity-ll 158 15.1 Introduction 158 15.2 Mixed hardening 163 15.3 Kinematic hardeningfor plane stress 164viii CONTENTS 15.4 Radial return with mixed linear hardening 166 15.5 Radial return with non-linear hardening 167 15.6 A general backward-Euler return with mixed linear hardening 168 15.7 A backward-Euler procedurefor plane stress with mixed linear hardening 170 15.8 A consistent tangent modular tensor following the radial return of Section 15.4 172 15.9 General form of the consistent tangent modular tensor 173 15.10 Overlay andother hardening models 174 15.1 0.1 Sophisticated overlay model 178 15.10.2 Relationship with conventional kinematic hardening 180 15.10.3 Other models 180 15.11 Computer exercises 181 15.12 Viscoplasticity 182 15.12.1 The consistenttangent matrix 184 15.12.2 Implementation 185 15.13Special notation 185 15.14 References 186 16 Largerotations 108 16.1 Non-vectorial large rotations 188 16.2 A rotation matrix for small (infinitesimal)rotations 188 16.3 A rotation matrix for large rotations (Rodrigues formula) 191 16.4 The exponential form for the rotationmatrix 194 16.5 Alternative forms for the rotation matrix 194 16.6 Approximations for the rotationmatrix 195 16.7 Compound rotations 195 16.8 Obtaining the pseudo-vectorfrom the rotation matrix, R 197 16.9 Quaternions and Euler parameters 198 16.10Obtaining the normalised quarternion from the rotation matrix 199 16.11 Additive and non-additive rotationincrements 200 16.12 The derivative of the rotation matrix 202 16.13 Rotating a triad so that one unit vector moves to a specified unit vector via the ‘smallest rotation’ 202 16.14 Curvature 204 16.14.1 Expressionsfor curvaturethat directly use nodal triads 204 16.14.2 Curvature without nodal triads 207 16.15 Special notation 211 16.16 References 212 17 Three-dimensional formulations for beams and rods 213 17.1 A co-rotationalframework for three-dimensional beam elements 213 17.1.1 Computing the local‘displacements’ 216 17.1.2 Computationof the matrix connecting the infinitesimallocal and global variables 218 17.1.3 The tangent stiffness matrix 221 17.1.4 Numerical implementationof the rotational updates 223 17.1.5 Overall solution strategywith a non-linear ‘local element’ formulation 223 17.1.6 Possible simplifications 225 17.2 An interpretation of an element due to Simo and Vu-Quoc 226 17.2.1 The finite element variables 227 17.2.2 Axial and shear strains 227 17.2.3 Curvature 228CONTENTS ix 17.2.4 Virtual work and the internalforce vector 229 17.2.5 The tangent stiffness matrix 229 17.2.6 An isoparametric formulation 231 17.3 An isoparametric Timoshenko beam approach using the total Lagrangian formulation 233 17.3.1 The tangent stiffness matrix 237 17.3.2 An outline of the relationship with the formulationof Dvorkinet al. 239 17.4 Symmetry and the use of different ‘rotation variables’ 240 17.4.1 A simple model showingsymmetry and non-symmetry 241 17.4.2 Using additive rotationcomponents 242 17.4.3 Considering symmetry at equilibriumfor the element of Section 17.2 243 17.4.4 Using additive (in the limit) rotationcomponents with the element of Section 17.2 245 17.5 Various forms of applied loading including ‘follower levels’ 248 17.5.1 Point loads appliedat a node 248 17.5.2 Concentratedmoments appliedat a node 249 17.5.3 Gravity loadingwith co-rotationalelements 251 17.6 Introducingjoints 252 17.7 Special notation 256 17.8 References 257 18 More on continuum and shell elements 260 18.1 Introduction 260 18.2 A co-rotationalapproachfor two-dimensional continua 262 18.3 A co-rotationalapproachfor three-dimensional continua 266 18.4 A co-rotational approachfor a curved membrane using facet triangles 269 18.5 A co-rotational approachfor a curved membrane using quadrilaterals 271 1 8.6 A co-rotational shell formulation with three rotational degrees of freedom per node 273 18.7 A co-rotationalfacet shellformulation basedon Morley’s triangle 276 18.8 A co-rotational shell formulation with two rotational degrees of freedom per node 280 18.9 A co-rotational frameworkfor the semi-loof shells 283 18.10An alternative co-rotational frameworkfor three-dimensionalbeams 285 18.1 0.1 Two-dimensionalbeams 286 18.1 1 Incompatible modes, enhanced strains and substitute strainsfor continuum elements 287 8.1 1.1 Incompatiblemodes 287 18.11.2 Enhanced strains 291 18.1 1.3 Substitute functions 293 18.1 1.4 Numerical comparisons 295 18.12 Introducing extra internal variables into the co-rotational formulation 296 18.13 Introducing extra internal variablesinto the Eulerian formulation 298 18.14 Introducing large elastic strains into the co-rotationalformulation 300 18.15 A simple stability test and alternative enhancedF formulations 301 18.16Special notation 304 18.17 References 305 19 Large strains and plasticity 308 19.1 Introduction 308 19.2 The multiplicativeF,F, approach 309X CONTENTS 19.3 Usingthe F,F, approachto arrive at the conventional ‘rate form’ 31 2 19.4 Usingthe rate form with an ‘explicit dynamic code’ 315 19.5 Integrating the rate equations 316 19.6 An F,F, update basedon the intermediate configuration 320 19.7 An F,F, update basedon the final (current) configuration 324 19.7.1 The flow rule 326 19.8 The consistent tangent 326 19.8.1 The limitingcase 327 19.9 Introducing large elasto-plastic strains into the finite element formulation 328 19.10 A simple example 332 19.11 Special notation 334 19.12 References 335 20 Stability theory 338 20.1 Introduction 338 20.2 General theory without ‘higher-order terms’ 338 20.2.1 Limit point 343 20.2.2 Bifurcation point 343 20.3 The introductionof higher-order terms 344 20.4 Classificationof singular points 346 20.4.1 Limit points 346 20.4.2 Bifurcation points 347 20.4.3 Symmetric bifurcations 347 20.4.4 Asymmetric bifurcations 347 20.5 Computationof higher-order derivativesfor truss elements 349 20.5.1 Amplificationof notation 349 20.5.2 Truss element usingGreen’s strain 350 20.5.3 Truss elements usinga rotated engineering strain 351 20.5.4 Computationof the stability coefficientsS,-S, 352 20.6 Special notation 352 20.7 References 353 21 Branch switching and further advanced solution procedures 354 21.1 Indirect computationof singular points 355 21.2 Simplebranchswitching 359 21.2.1 Corrector basedon a linearised arc-length method 360 21.2.2 Corrector using displacement control at a specified variable 361 21.2.3 Corrector usinga ‘cylindricalarc-lengthmethod’ 361 21.3 Branch switching using higher-order derivatives 36 1 21.4 General predictorsusing higher-order derivatives 362 21.4.1 Loadcontrol 363 21.4.2 Displacement control at a specified variable 363 21.4.3 The ‘cylindricalarc-lengthmethod’ 364 21.5 Correctorsusinghigher-order derivatives 365 21.6 Direct computationof the singular points 366 21.7 Line-searcheswith arc-length and similar methods 368 21.7.1 Line-searches with the RiksMlempnerarc-length method 368 21.7.2 Line-searches with the cylindrical arc-length method 370 21.7.3 Uphillor downhill? 373 21.8 Alternativearc-lengthmethods using relative variables 373 21.9 An alternative methodfor choosing the root for the cylindrical arc-lengthmethod 374CONTENTS xi 21.10 Statiddynamicsolution procedures 376 21.1 1 Special notation(seealso Section 20.6) 378 21.12 References 379 22 Examplesfrom an updated non-linear finite element computer program using truss elements (written in conjunctionwith Dr Jun Shi) 381 22.1 A two-bar truss with an asymmetric bifurcation 382 22.1.1 Bracketing 22.1.2 Branch switching 22.2 The von Misestruss 22.2.1 Bracketing 22.2.2 Branch switching 22.3 A three-dimensional dome 22.3.1 Bracketing 22.3.2 Branch switching 22.3.3 The higher-order predictor 22.3.4 The higher-order correctors 22.3.5 Line searches 22.4 A three-dimensional arch truss 22.5 A two-dimensional circular arch 22.6 References 23 Contactwith friction 23.1 Introduction 23.2 A two-dimensionalfrictionlesscontact formulation using a penalty approach 412 23.2.1 Some modifications 415 23.3 The ‘contact patchtest’ 417 23.4 Introducing ‘sticking friction’ in two dimensions 420 23.5 IntroducingCoulomb ‘sliding friction’ in two dimensions 422 23.6 Using Lagrangian multipliers insteadof the penalty approach 424 23.7 The augmented Lagrangian methods 426 23.8 An augmented Lagrangian technique with Coulomb ’sliding friction’ 429 23.8.1 A symmetrised version 430 23.9 A three-dimensional frictionless contact formulationusinga penalty approach 431 23.9.1 The consistent tangent matrix 434 23.10 Adding ‘sticking friction’ in three dimensions 435 23.10.1 The consistent tangent matrix 437 23.11 Coulomb ‘sliding friction’ in three dimensions 430 23.12 A penalty/barrier methodfor contact 439 23.1 3 Amendments to the solution procedures 441 24.14 Special notation 442 23.15 References 444 24 Non-linear dynamics 447 24.1 Introduction 447 24.2 Newmark’s method 447 24.3 The ‘average acceleration method’or ‘trapezoidalrule’ 448 24.4 The ‘implicit solution procedure’ 448xii CONTENTS 24.4.1 The ‘predictorstep’ 449 24.2.2 The ‘corrector’ 449 24.5 An explicit solution procedure 450 24.6 A staggered, central difference, explicit solutionprocedure 451 24.7 Stability 452 24.8 The Hilber-Hughes-Taylor s( method 455 24.9 More on the dynamic equilibriumequations 456 24.10 An energy conserving total Lagrangian formulation 458 24.10.1 The ‘predictor step’ 460 24.10.2 The ‘corrector’ 460 24.11 A co-rotational energy-conserving procedurefor two-dimensionalbeams 461 24.11.1 Sophistications 463 24.11.2 Numericalsolution 464 24.12 An alternative energy-conservingprocedure for two-dimensionalbeams 466 24.13 Automatic time-stepping 468 24.14 Dynamic equilibriumwith rotations 470 24.15 An ‘explicit co-rotational procedure’for beams 473 24.16 Updating the rotational velocities and accelerations 474 24.17 A simple implicit co-rotational procedure using rotations 476 24.18 An isoparametric formulationfor three-dimensionalbeams 477 24.19 An alternative implicit co-rotational formulation 479 24.20 (Approximately) energy-conserving co-rotational procedures 480 24.21 Energy-conserving isoparametric formulation 483 24.22 Special notation 485 24.23 References 486 Index
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