Akbarov Stability Loss and Buckling Delamination

Three-Dimensional Linearized Approach for Elastic and Viscoelastic Composites
1. Auflage 2012
ISBN: 978-3-642-30290-9
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 450 Seiten

Reihe: Engineering (R0)

This book investigates stability loss problems of the viscoelastic composite materials and structural members within the framework of the Three-Dimensional Linearized Theory of Stability (TDLTS). The stability loss problems are considered the development of the initial infinitesimal imperfection in the structure of the material or of the structural members. This development is studied within the framework of the Three-Dimensional Geometrical Non-Linear Theory of the Deformable Solid Body Mechanics. The solution to the corresponding boundary-value problems is presented in the series form in the small parameter which characterizes the degree of the initial imperfection. In this way, the nonlinear problems for the domains bounded by noncanonical surfaces are reduced for the same nonlinear problem for the corresponding domains bounded by canonical surfaces and the series subsequent linearized problems. It is proven that the equations and relations of these linearized problems coincide with the corresponding ones of the well-known TDLTS. Under concrete investigations as stability loss criterion the case is taken for the initial infinitesimal imperfection that starts to increase indefinitely. Moreover, it is proven that the critical parameters can be determined by the use of only the zeroth and first approximations.

Autoren/Hrsg.

Akbarov, Surkay

Inhaltsverzeichnis

1;Stability Loss and Buckling Delamination;2
1.1;Preface;5
1.2;Acknowledgments;6
1.3;Contents;7
1.4;Abbreviations;12
1.5;1 Introduction;13
1.5.1;1.1…A General Background;13
1.5.2;1.2…Text Preview;17
1.5.3;References;19
1.6;2 Stability Loss ProblemsProblems Related to SolidSolid and HollowHollow CircularCircular CylinderCircular cylinders;21
1.6.1;2.1…Formulation of the Problem Related to the Global Stability Loss ;21
1.6.2;2.2…Method of SolutionMethod of solution for the Global Stability Lossstability Loss Problem ;25
1.6.3;2.3…Approximate Equations for the Stability LossStability loss of the CylinderCylinder-Beam Obtained from Equations of the TDLTS by the Average-IntegratingAverage-Integrating ProcedureAverage-integrating procedure;38
1.6.3.1;2.3.1 Bernoulli BeamBernoulli Beam theory;38
1.6.4;2.4…The Third Order Refined Beam TheoryThird Order Refined Beam Theory;42
1.6.5;2.5… Numerical Results and Discussions;44
1.6.5.1;2.5.1 Solid CylinderCylinder;47
1.6.5.2;2.5.2 Hollow CylinderCylinder;53
1.6.6;2.6…Formulation of the ProblemProblem Related to the Rotationally SymmetricRotationally symmetric Locallocal Stability LossLocal stability loss ;57
1.6.7;2.7…Method of SolutionMethod of solution for the Rotationally SymmetricRotationally symmetric ProblemRotationally symmetric problem;60
1.6.8;2.8…Approximate Equations of the Stability Loss of the Cylinder-Shell Obtained from Equations of the TDLTS by the Average-Integrating Procedure;67
1.6.8.1;2.8.1 Kirchhoff--Love Shell Theory;67
1.6.9;2.9…The Third Order Refined Shell TheoryRefined shell theory;71
1.6.10;2.10…Numerical Results Related to the Rotationally Symmetric Stability Loss Problemproblem;73
1.6.11;2.11…Conclusions;80
1.6.12;References;81
1.7;3 Stability Loss ProblemsProblems for ViscoelasticViscoelastic Plates;83
1.7.1;3.1…Formulation of the ProblemProblem and Basic Field EquationsBasic field equations;83
1.7.2;3.2…Approach for the Solution to the Stability LossStability loss ProblemProblem for RectangularRectangular PlatePlate;86
1.7.3;3.3…Simply Supported RectangularRectangular PlatePlate;92
1.7.3.1;3.3.1 Deriving Approximate Equations of the Stability LossStability loss of the Simply SupportedSimply supported RectangularRectangular PlatePlate from Equations of the TDLTS by the Average-IntegratingAverage-integrating Procedure;92
1.7.3.1.1;3.3.1.1 Kirchhoff-Love Plate TheoryKirchhoff-Love Plate Theory;93
1.7.3.1.2;3.3.1.2 The Third Order Refined PlatePlate Theory;97
1.7.3.2;3.3.2 Solution for the Formulated Mathematical ProblemsProblems for the Simply SupportedSimply supported PlatePlate;100
1.7.3.3;3.3.3 Numerical Results and Discussions;102
1.7.4;3.4…Rectangular PlatePlate ClampedClamped at Two Opposite Ends and Simply Supportedsimply supported at the Two Other Opposite Ends;106
1.7.4.1;3.4.1 Solution Procedure of the Equations of the TDLTS. Semi-Analytical FEM Modeling;106
1.7.4.2;3.4.2 Solution Procedure for the Approximate Stability LossStability loss Equations;110
1.7.4.3;3.4.3 Numerical Results and Discussions;111
1.7.5;3.5…Rectangular PlatePlate ClampedClamped at all Ends;114
1.7.5.1;3.5.1 Solution Procedure for the TDLTS ProblemProblem. 3D FEM Modeling;114
1.7.5.2;3.5.2 Solution Procedure for the Approximate Stability LossStability loss ProblemsProblems;116
1.7.5.3;3.5.3 Numerical Results and Discussions;117
1.7.6;3.6…Symmetric Stability LossStability loss of the CircularCircular PlateCircular plate;119
1.7.6.1;3.6.1 Formulation of the ProblemProblem and Governing Field EquationsField equations;119
1.7.6.2;3.6.2 Method of SolutionMethod of Solution;122
1.7.6.2.1;3.6.2.1 Approximate Loss of Stability Equations of a CircularCircular PlateCircular plate with Initial ImperfectionInitial imperfection;124
1.7.6.2.2;3.6.2.2 Solution Procedure for the TDLTS ProblemsProblems;125
1.7.6.2.3;3.6.2.3 Solution Procedure for the Approximate Equations of Stability;128
1.7.6.3;3.6.3 Numerical Results and Discussions;129
1.7.6.3.1;3.6.3.1 Comparison of Some Numerical Results Obtained in the Framework of the TDLTS for the Pure ElasticPure elastic PlatePlate;131
1.7.6.3.2;3.6.3.2 Comparison of the Results Obtained for the Pure Elasticpure elastic PlatePlate in the Framework of the TDLTS and Approximate TheoriesTheories;133
1.7.6.3.3;3.6.3.3 Viscoelastic Plateplate;134
1.7.7;3.7…Stability Loss of the RotatingRotating CircularCircular and AnnularAnnular Discs;137
1.7.7.1;3.7.1 Formulation of the ProblemProblem;137
1.7.7.2;3.7.2 Method of SolutionMethod of Solution;139
1.7.7.3;3.7.3 Numerical Results and Discussions;141
1.7.8;References;144
1.8;4 Buckling Delamination of Elastic and ViscoelasticViscoelastic CompositeComposite Plates with Cracks;146
1.8.1;4.1…Background of Related ProblemsProblems;146
1.8.2;4.2…Buckling Delamination ProblemsProblems for PlatePlate-Strips with a Single CrackCrack;148
1.8.2.1;4.2.1 Formulation of the ProblemsProblems;148
1.8.2.2;4.2.2 Method of SolutionMethod of Solution;151
1.8.2.2.1;4.2.2.1 Determination of the Zeroth ApproximationZeroth Approximation;154
1.8.2.2.2;4.2.2.2 Determination of the First ApproximationFirst Approximation;156
1.8.2.3;4.2.3 Numerical Results and Discussions;157
1.8.2.3.1;4.2.3.1 Testing of the Algorithm Used for Obtaining Numerical Results;157
1.8.2.3.2;4.2.3.2 Buckling Delamination of a Simply SupportedSimply Supported Elastic PlatePlate-Strip;158
1.8.2.3.3;4.2.3.3 Buckling Delamination of the ClampedClamped Elastic Plateplate-Strip;160
1.8.2.3.4;4.2.3.4 Buckling Delamination of the ClampedClamped ViscoelasticViscoelastic PlatePlate-Strip;162
1.8.3;4.3…Buckling Delamination of the PlatePlate-Strip with Two Parallel CracksTwo Parallel Cracks;164
1.8.3.1;4.3.1 Mathematical Formulation of the ProblemProblem;164
1.8.3.2;4.3.2 Method of SolutionMethod of Solution: FEM Modeling;166
1.8.3.3;4.3.3 Numerical Results and Discussions;167
1.8.4;4.4…Buckling Delamination of the PlatePlate-Strip with Two Collinear CracksTwo Collinear Cracks;170
1.8.4.1;4.4.1 Formulation of the ProblemProblem and Solution Method;170
1.8.4.2;4.4.2 Numerical Results and Discussions;172
1.8.5;4.5…Buckling Delamination of the Three-LayeredLayered (SandwichSandwich) PlatePlate-Strip with Two Parallel InterfaceInterface Cracks;175
1.8.5.1;4.5.1 Formulation of the ProblemProblem;175
1.8.5.2;4.5.2 Method of SolutionMethod of Solution;177
1.8.5.3;4.5.3 Numerical Results and Discussions;183
1.8.6;4.6…Buckling Delamination of the Three-LayeredLayered (SandwichSandwich) PlatePlate-Strip with Two Collinear Interface CracksTwo Collinear Interface Cracks;187
1.8.6.1;4.6.1 Formulation of the ProblemProblem and Method of Solution;187
1.8.6.2;4.6.2 Numerical Results and Discussions;190
1.8.7;4.7…Buckling Delamination of the Elastic and Viscoelstic CompositeComposite CircularCircular PlateCircular Plate-Disc with a Penny-Shaped CrackPenny-Shaped Crack;192
1.8.7.1;4.7.1 Formulation of the ProblemProblem;192
1.8.7.2;4.7.2 Method of SolutionMethod of Solution;194
1.8.7.2.1;4.7.2.1 The Zeroth ApproximationZeroth Approximation;195
1.8.7.2.2;4.7.2.2 The First ApproximationFirst Approximation;196
1.8.7.3;4.7.3 Numerical Results and Discussions;197
1.8.8;4.8…Buckling Delamination of the Three-LayeredLayered (SandwichSandwich) CircularCircular PlateCircular Plate-Disc with Two Parallel Interface Penny-Shaped CracksTwo Parallel Interface Penny-Shaped Cracks;201
1.8.8.1;4.8.1 Formulation of the ProblemProblem;201
1.8.8.2;4.8.2 Method of Solution;204
1.8.8.2.1;4.8.2.1 The Zeroth ApproximationZeroth Approximation;208
1.8.8.2.2;4.8.2.2 The First ApproximationFirst Approximation;209
1.8.8.3;4.8.3 Numerical Results and Discussions;211
1.8.9;4.9…Remarks on the FEM Modeling of the CrackCrack’s Tips;213
1.8.10;4.10…Buckling Delamination of a RectangularRectangular PlatePlate Containing a Rectangular CrackCrack;216
1.8.10.1;4.10.1 Formulation of the ProblemsProblems;217
1.8.10.2;4.10.2 Solution Method;220
1.8.10.3;4.10.3 FEM Modeling;223
1.8.10.4;4.10.4 Numerical Results and Discussions;224
1.8.10.4.1;4.10.4.1 Numerical Results Obtained in Case 1 (BandBand CrackBand Crack);225
1.8.10.4.2;4.10.4.2 Numerical Results Obtained in Case 2 (EdgeEdge CrackCrack);231
1.8.10.4.3;4.10.4.3 Numerical Results Obtained in Case 3 (Embedded CrackCrack);236
1.8.10.4.4;4.10.4.4 Main Conclusions Based on the Numerical Results;240
1.8.11;4.11…Buckling Delamination of a SandwichSandwich RectangularRectangular PlatePlate with InterfaceInterface Rectangular Cracks;242
1.8.11.1;4.11.1 Formulation of the ProblemProblem;242
1.8.11.2;4.11.2 Solution Method;248
1.8.11.3;4.11.3 FEM Modeling;252
1.8.11.4;4.11.4 Numerical Results and Discussions;253
1.8.11.4.1;4.11.4.1 Numerical Results Obtained in Case 1 (BandBand CrackBand Cracks);254
1.8.11.4.2;4.11.4.2 Numerical Results Obtained in Case 2 (EdgeEdge Cracks);258
1.8.11.4.3;4.11.4.3 Numerical Results Obtained in Case 3 (Embedded Cracks);265
1.8.11.4.4;4.11.4.4 Conclusions Based on the Numerical Results;273
1.8.12;References;276
1.9;5 Surface and InternalInternal Stability LossInternal Stability Loss in the Structure of Elastic and ViscoelasticViscoelastic LayeredLayered CompositesLayered Composites;279
1.9.1;5.1…Background of Related ProblemsProblems;279
1.9.2;5.2…Stability Loss in the Structure of Elastic and ViscoelasticViscoelastic LayeredLayered CompositesLayered Composites with PeriodicalPeriodical Initial ImperfectionsInitial Imperfection;281
1.9.2.1;5.2.1 Formulation of the ProblemProblem on the Determination of the Stress--Strain StateStress--Strain State in a LayeredLayered CompositeComposite with an Arbitrary Number of Layers with Initially Infinitesimal Imperfections;281
1.9.2.2;5.2.2 Method of SolutionMethod of Solution;283
1.9.2.2.1;5.2.2.1 The Equations of Contact SurfacesContact Surfaces;283
1.9.2.2.2;5.2.2.2 The Employing the BoundaryBoundary formForm Perturbation Technique;285
1.9.2.2.3;5.2.2.3 Plane-Strain StatePlane-Strain State in Case 1 (Co-PhaseCo-Phase ImperfectionsCo-Phase Imperfections of Layers);288
1.9.2.2.4;5.2.2.4 Plane-Strain StatePlane-Strain State in Case 2 (Imperfection of a Single LayerImperfection of a Single Layer);296
1.9.2.3;5.2.3 Numerical Results and Discussions;298
1.9.2.3.1;5.2.3.1 Numerical Results Related to Case1 (Co-PhaseCo-Phase CurvingCo-Phase Curving of LayersCo-Phase Curving of Layers);298
1.9.2.3.2;5.2.3.2 Numerical Results Related to Case 2 (Imperfection of a Single LayerImperfection of a Single Layer);300
1.9.2.3.3;5.2.3.3 Numerical Results Related to ViscoelasticViscoelastic CompositeComposite Materials;302
1.9.3;5.3…Stability Loss in the Structure of the Elastic and ViscoelasticViscoelastic LayeredLayered CompositesLayered Composites with LocalLocal Initial ImperfectionsLocal Initial Imperfections;304
1.9.3.1;5.3.1 Formulation of the ProblemProblem and Method of Solution;304
1.9.3.2;5.3.2 Numerical Results and Discussions;306
1.9.4;5.4…The Influence of the Inclination of the LocalLocal Initial ImperfectionsLocal Initial Imperfections of the Reinforcing LayersReinforcing Layers on the Values of the CriticalCritical Parameters;312
1.9.4.1;5.4.1 Formulation of the ProblemProblem and Solution Method;312
1.9.4.2;5.4.2 Results and Discussions;316
1.9.4.3;5.4.3 Conclusions;318
1.9.5;5.5…Surface Undulation InstabilitySurface Undulation Instability of the ViscoelasticViscoelastic Half-Space Covered with a Stack of Layers in Bi-Axial Compression;319
1.9.5.1;5.5.1 Formulation of the ProblemProblem;320
1.9.5.2;5.5.2 Method of SolutionMethod of Solution;322
1.9.5.3;5.5.3 Numerical Results and Discussions;329
1.9.5.3.1;5.5.3.1 Two-Dimensional ProblemsProblems;329
1.9.5.3.2;5.5.3.2 Three-Dimensional ProblemsProblems;336
1.10;6 Stability Loss in the Structure of Unidirected FibrousFibrous Elastic and ViscoelasticViscoelastic CompositesComposites;346
1.10.1;6.1…Some General Remarks on the Field EquationsField equations, ProblemProblem Formulations and Method of Solution;346
1.10.1.1;6.1.1 General Remarks on the Field EquationsField equations and ProblemProblem Formulations;346
1.10.1.2;6.1.2 General Remarks on the Method of Solution;350
1.10.2;6.2…Micro BucklingMicro Buckling of a Single Fiber in the ViscoelasticViscoelastic Matrix;357
1.10.2.1;6.2.1 Formulation of the ProblemProblem and Method of Solution;357
1.10.2.2;6.2.2 Numerical Results and Discussions;359
1.10.3;6.3…Internal Stability LossStability loss of Two Neighboring FibersTwo neighboring fibers in a ViscoelasticViscoelastic Matrix;362
1.10.3.1;6.3.1 Formulation of the ProblemProblem and Method of Solution;362
1.10.3.2;6.3.2 Numerical Results and Discussion;371
1.10.4;6.4…Internal Stability LossStability loss of a Row of Unidirected Periodically Located Fibersrow of Unidirected Periodically Located fibers in a ViscoelasticViscoelastic Matrix;377
1.10.4.1;6.4.1 Formulation of the ProblemProblem and Solution Method;377
1.10.4.2;6.4.2 Numerical Results and Discussions;382
1.10.5;6.5…Stability Loss of a Micro-Fiber in an Elastic and a ViscoelasticViscoelastic Matrix Near the Free Convex Cylindrical SurfaceConvex cylindrical surface;383
1.10.5.1;6.5.1 Formulation of the ProblemProblem;383
1.10.5.2;6.5.2 Method of SolutionMethod of solution;386
1.10.5.3;6.5.3 Numerical Results and Discussions;400
1.10.6;References;408
1.11;Supplement 1: Applications of the Approach Developed in Chap. 4 on the Problems Related to the Stress Concentration in Initially Stressed Bodies;410
1.11.1;S1.1 The Influence of the Initial Stresses on the SIFand ERR at Crack Tips in a Plate-Stripfrom Orthotropic Material;410
1.11.1.1;Conclusions;420
1.11.1.2;S1.1.1 Formulation of the Problem;411
1.11.1.3;S1.1.2 FEM Modeling;413
1.11.1.4;S1.1.3 Numerical Results and Discussion;415
1.11.2;S1.2 The Influence of the Initial Tension of a Stripwith a Rectangular Hole on the Stress ConcentrationCaused by Additional Loading;421
1.11.2.1;S1.2.1 Formulation of the Problem;422
1.11.2.2;S1.2.2 FEM Modeling;423
1.11.2.3;S1.2.3 Numerical Results and Discussions;426
1.11.2.4;Conclusions;431
1.12;Supplement 2: Self-Balanced StressesCaused by Periodical Curving of TwoNeighboring and Periodically LocatedRow of Fibers in an Infinite Matrix;433
1.12.1;S2.1 Numerical Results Related to the Case Where an InfiniteMatrix Contains Two Neighboring Fibers;433
1.12.2;S2.2 Numerical Results Related to the Case Wherean Infinite Matrix Contains PeriodicallyLocated Row of Fibers;438
1.13;Index;448