Recent Developments in the Theory of Shells (eBook)
798 Seiten
Springer International Publishing (Verlag)
978-3-030-17747-8 (ISBN)
This book commemorates the 80th birthday of Prof. W. Pietraszkiewicz, a prominent specialist in the field of general shell theory. Reflecting Prof. Pietraszkiewicz's focus, the respective papers address a range of current problems in the theory of shells. In addition, they present other structural mechanics problems involving dimension-reduced models. Lastly, several applications are discussed, including material models for such dimension-reduced structures.
Prof. Dr.-Ing.habil.Dr.h.c.mult Holm Altenbach, Member of the International Association of Applied Mathematics and Mechanics, and the International Research Center on Mathematics and Mechanics of Complex Systems (M&MoCS), Italy. Employment history includes positions at Otto-von-Guericke-University Magdeburg and at Martin Luther University Halle-Wittenberg, both Germany. Graduated from Leningrad Polytechnic Institute in 1985 (diploma in Dynamics and Strength of Machines). Defended PhD in 1983, awarded Doctor of technical sciences in 1987, both at the same Institute.
Present position: Full Professor in Engineering Mechanics at the Otto-von-Guericke-University, Faculty of Mechanical Engineering, Institute of Mechanics (since 2011), acting director of the Institute of Mechanics since 2015Areas of scientific interest:
• General theory of elastic and inelastic plates and shells.
• Creep and damage mechanics.
• Strength theories.
• Nano- and micromechanics.
Author/Co-author/Editor of 60 Books (textbooks/monographs/proceedings), appr. 380 scientific papers (among them 250 peer-reviewed) and 500 scientific lectures. Managing Editor (2004 to 2014) and Editor-in-Chief (2005 - to date) of the Journal of Applied Mathematics and Mechanics (ZAMM) - the oldest journal in Mechanics in Germany (founded by Richard von Mises in 1921), Advisory Editor of the journal 'Continuum Mechanics and Thermodynamics' since 2011, Associate Editor of the journal 'Mechanics of Composites' (Riga) since 2014, Co-Editor of the Springer Series 'Advanced Structured Materials' since 2010.
Awards: 1992 Krupp-Award (Alexander von Humboldt-Foundation), 2000 Best paper of the year Journal of Strain Analysis for Engineering Design, 2003 Gold Medal of the Faculty of Mechanical Engineering, Politechnika Lubelska, Lublin, Poland, 2004 Semko-Medal of the National Technical University Kharkov, Ukraine, 2007 Doctor honoris causa, National Technical University Kharkov, Ukraine, 2011 Fellow of the Japanese Society for the Promotion of Science, 2014 Doctor honoris causa, University Constanta, Romania, 2016 Doctor honoris causa, Vekua-Institute, Tbilisi, Georgia, 2018 Alexander von Humboldt Award (Poland)
Areas of scientific interest:
• Theory of plates and shells with applications.
• Continuum mechanics.
• Generalized media.
• Nonlinear elasticity. • Nano- and micromechanics.Author/Editor of 12 books written in Russian/English/Spanish, author/co-author of ca. 150 scientific peer-reviewed papers indexed by Scopus/Web of Science. Member of editorial board of the following journals: Journal of Applied Mathematics and Mechanics (ZAMM); Mathematics and Mechanics of Solids; Acta Mechanica; Technische Mechanik; Nanoscience and Technology: An International Journal; PNRPU Mechanics Bulletin.
Award: International Prize 'Tullio Levi-Civita' for the Mathematical and Mechanical Sciences - 2018.
Prof. dr hab. in?. Jacek Chró?cielewski, Member of Polish Society of Theoretical and Applied Mechanics; Polish Association for Computational Mechanics; Polish Society of Bridge Engineers. Between 1984-2004 he spent a total of 5 years working at Ruhr-Universität Bochum, Bergische Universität Wuppertal and RWTH Aachen, occupying various scientific positions. Currently is working at Faculty of Civil and Environmental Engineering, Gda?sk University of Technology, Gda?sk, Poland.
Areas of scientific interest:
• Theory of plates and shells with applications
• Finite elements method
• Biomechanics
• Composites
• Bridge structures
• Road safety elements
He is an author and co-author of 2 books written in Polish on theory and finite element analysis of shells. He is an author and co-authored of numerous scientific and practical papers of with 56 indexed in Web of Science. Prof. Chró?cielewski is a known expert in the field of bridge structures, he performed more than 300 expertise's in this field.
Most important scientific awards:
Professor's W. Olszak Scientific Prize of Polish Society of Theoretical and Applied Mechanics,
Professor's O.C. Zienkiewicz Medal of Polish Association for Computational Mechanics,
Professor's J. Szmelter Prize of Military University of Technology, Warsaw, Poland.
Preface 7
Selected publications of Prof. Wojciech Pietraszkiewicz 11
Books 11
Edited Books 12
Selected Papers 12
Contents 18
Tribute to Professor Wojciech Pietraszkiewicz 22
References 24
Computer Modeling of Nonlinear Deformation and Loss of Stability of Composite Shell Structures Under a Combined Effect of Quasi-static and Pulsed Loads 25
1 Introduction 26
2 Constructing the Resolving Equation Set 27
3 The Numerical Method of Analyzing the Initial Boundary-Value Problem 32
4 Investigation Results 35
4.1 Numerical Analysis of Loss of Stability of Cylindrical Shells, Accounting for Quasistatic Preloading 35
4.2 Numerical Analysis of Loss of Stability of Isotropic and Composite Cylindrical Shells Under Dynamic External Pressure and (or) Axial Compression 42
4.3 Numerical Analysis of Dynamic Buckling of Discretely Stiffened Cylindrical Shells Loaded by Axial Compression and External Pressure 46
5 Conclusion 50
References 51
Analytical Buckling Analysis of Cylindrical Shells with Elliptic Cross Section Subjected to External Pressure 53
1 Introduction 53
2 Characteristics of the Elliptic Cross Section 54
3 Semimembrane Theory 55
4 Perturbation Solution of Eigenvalue Problem 56
5 Proof of Accuracy 58
6 Concluding Remarks 59
References 59
Subclasses of Mechanical Problems Arising from the Direct Approach for Homogeneous Plates 62
1 Introduction 62
1.1 Motivation 62
1.2 Frame of Reference 64
1.3 Preliminaries 66
2 The Original Problem 67
2.1 Kinematics 68
2.2 Kinetics 68
2.3 Equilibria 69
2.4 Boundary Conditions 70
2.5 Constitutive Relations 71
2.6 Variational Principle 73
2.7 Classification and Formalization 74
3 Subclasses 75
3.1 In-Plane Loaded Plate Problem 75
3.2 Out-of-Plane Loaded Plate Problems 76
4 Conclusion 79
References 80
Large Oscillations Around Curled Equilibrium Configurations of Uniformly Loaded Euler–Bernoulli Beams: Numerical and Experimental Evidences 83
1 Introduction 83
2 Naturally Discrete Model of Elastica 85
3 Physical Experiments and Numerical Simulations 87
3.1 Using the Curled Static Configuration to Estimate Mechanical Parameters 88
3.2 Large Oscillations Around Curled Stable Equilibrium Configurations 89
4 Concluding Remarks and Future Challenges 92
References 93
Unsymmetrical Wrinkling of Nonuniform Annular Plates and Spherical Caps Under Internal Pressure 97
1 Introduction 97
2 Problem Formulation 98
2.1 Axisymmetric Behavior 101
3 Equations for Buckling 101
4 Numerical Results and Discussion 102
5 Conclusion 106
References 107
Two-Dimensional Model of a Plate, Made of Material with the General Anisotropy 108
1 Introduction 109
2 The Main Equations and Assumptions 110
3 Transformation of System (2.12) 112
4 Asymptotic Solution of Eqs.(3.4). The Zero Approximation 113
5 The Higher Approximations 115
6 Two Examples of Elasticity Relations 117
7 Harmonic Solution 120
8 Harmonic Vibrations 122
9 Conclusions 124
References 124
An Alternative Approach to the Buckling Resistance Assessment of Steel, Pressurised Spherical Shells 126
1 Introduction 127
2 Elastic Critical Buckling Resistance and Plastic Reference Resistance 128
3 Modified Capacity Curves—Buckling Parameters 130
4 Buckling Capacity Curves 134
5 Determination of the Buckling Resistance—The Flow Chart 138
6 Illustrative Examples 139
6.1 Calculation Example No. 1 139
6.2 Calculation Example No. 2 142
7 Comparison with Experimental Results 146
7.1 Experimental Research by Kaplan and Fung [10] 146
7.2 Experimental Research by Seaman [17] 147
7.3 Experimental Research by B?achut [1] 149
8 Conclusions 150
References 151
Asymptotically-Accurate Nonlinear Hyperelastic Shell Constitutive Model Using Variational Asymptotic Method 152
1 Introduction 152
2 Analysis 153
2.1 Kinematics 154
2.2 Potential Due to Load 155
2.3 Total Potential Energy 156
2.4 Zeroth-Order Approximation 156
2.5 First-Order Approximation 157
2.6 Shell Stiffness Matrix 159
3 Nonlinear Shell Theory 159
3.1 Nonlinear Finite Element Analysis 160
4 Numerical Results 160
5 Conclusions 163
References 172
Three-Dimensional Finite Element Modelling of Free Vibrations of Functionally Graded Sandwich Panels 174
1 Introduction 175
2 Problem Statement 177
2.1 Equations of Motion 177
2.2 Constitutive Equations and FGM 178
2.3 Finite Element Solution Procedure 180
3 Three-Dimensional Graded Finite Element 181
4 Numerical Results 183
4.1 Static Analysis 183
4.2 Free Vibration Analysis 185
5 Conclusions 191
References 192
Recent Achievements in Constitutive Equations of Laminates and Functionally Graded Structures Formulated in the Resultant Nonlinear Shell Theory 195
1 Introduction 195
2 Failure Analysis of Multilayered Shells 197
2.1 Introduction 197
2.2 Progressive Failure Analysis 198
2.3 Numerical Example 200
2.4 Conclusions 201
3 Elastoplastic Analysis of FGM Shells 203
3.1 Introduction 203
3.2 Elastoplastic Constitutive Relation 203
3.3 Numerical Examples 207
4 Conclusions 213
References 213
On Optimal Archgrids 218
1 Introduction 218
2 Optimal Pin-Supported Planar Arches—The Direct Construction 220
3 The Variational Construction of the Optimal Arch 223
4 Optimal Archgrids 227
4.1 Setting of the Optimization Problem 227
4.2 Analysis of the Optimization Problem 230
5 Optimal Roof for a Uniform Vertical Load over a Square Domain 235
5.1 Solution to the Problem (78) 236
5.2 Solution to the Problem (75) 237
6 Final Remarks 238
References 239
Cylindrical Shell Model of Helical Type Wire Structures Accounting for Layers' Interaction 241
1 Introduction 241
2 Reduction of a Helix Layer to An Equivalent Cylindrical Shell 243
3 Stiffness of a Conductor 247
4 Modeling of Aluminum-Steel Conductors 251
5 Modeling of Spiral Reinforcements 253
6 Extension of the Connecting Clamp 254
6.1 The External and Conductive Layers are Deformed as a Whole 255
6.2 The External Layer Slides on the Conductive Layer 258
7 Example of a Connecting Clamp 260
8 Conclusion 262
References 262
Buckling of Cylindrical Shell Stiffened by Annular Plate Under External Pressure 264
1 Introduction 264
2 Basic Equations 266
3 Buckling of the First Type 267
3.1 Analysis of Boundary Conditions 267
3.2 The Asymptotic Approach for a Shell with Freely Supported Edge 269
3.3 Approximate Solutions in the Case of Generalized Boundary Conditions 272
3.4 Example 274
4 Buckling of the Second Type 275
4.1 Asymptotic Analysis of Boundary Conditions 275
4.2 Pre-buckling Stress-Resultants 277
4.3 Buckling of the Plate 279
5 Conclusions 282
References 282
2D Theory of Shell-like Tensegrity Structures 284
1 Introduction 284
2 The Concept of Tensegrity 285
3 Mechanical Properties of Tensegrity Shell-like Structures 287
4 Six-Parameter Shell Theory 289
5 Parametric Study of Cylindrical Tensegrity-like Shell 290
6 Conclusion 295
References 295
On Some Recent Discrete-Continuum Approaches to the Solution of Shell Problems 297
1 Introduction 297
2 Discrete Fourier Series Method in Static Problems for Thick-Walled Shells 298
2.1 Problem Statement 299
2.2 Problem-Solving Method 301
2.3 Analysis of the Numerical Results 303
3 Spline-Collocation Method 305
3.1 Problem Statement. Basic Equations 306
3.2 Problem-Solving Technique 310
3.3 Analysis of the Numerical Results 313
4 Complete Systems Method in Static Problems for Thin-Walled Shells 316
4.1 Problem Statement 316
4.2 Solving Method 317
4.3 Calculation Characteristics of the Method 321
4.4 Employment of the CSM in a Problem on Shell Vibrations 322
5 Conclusions 323
References 324
A Composite Wave Model for a Cylindrical Shell 326
1 Introduction 326
2 Statement of the Problem 327
3 Flat Plate 330
4 Numerical Results for Transverse Compression 331
5 Cylindrical Shell 333
6 Concluding Remarks 335
References 338
A Beam—Just a Beam in Linear Plane Bending 340
1 Introduction 340
2 Consistent Approximation 344
2.1 Taylor-Series Expansion 344
2.2 Uniform-Approximation Method 346
2.3 Pseudo-Reduction Technique 347
2.4 Stress and Load Resultants, Equilibrium Conditions 347
3 First-Order Approximation 349
3.1 ODE System 349
3.2 Zeroth-Order Approximation 349
3.3 Pseudo Reduction 350
3.4 Stress Resultants and Boundary Conditions 352
3.5 Extended Deformation Ansatz 353
4 Second-Order Approximation 355
5 Concluding Remarks 359
References 360
Inflation of a Cylindrical Membrane Partially Stretched over a Rigid Cylinder 362
1 Introduction 362
2 Equilibrium Equations 364
2.1 Basic Equations 364
2.2 Contact Area 366
2.3 Inflation Area 367
2.4 Conjugation Conditions 367
3 Bartenev–Khazanovich (Varga) Material 369
4 Results 370
5 Conclusion 374
References 374
Singular Surface Curves in the Resultant Thermodynamics of Shells 377
1 Introduction 377
2 Resultant Thermodynamics of Shells with Singular Curves 379
3 Continuity Conditions along Junctions 381
4 On Constitutive Equations in Six-Parameter Shell Theory 384
5 On Phase Transitions in Shells 386
6 Conclusions 387
References 388
Hybrid-Mixed Shell Finite Elements and Implicit Dynamic Schemes for Shell Post-buckling 392
1 Introduction 392
2 Hybrid-Mixed Shell Finite Elements 394
2.1 Geometrically Exact Shell Model 394
2.2 Shell Director Description 395
2.3 Rotation-Less Shell Model 396
2.4 Hybrid-Mixed Quadrilateral Finite Elements 397
2.5 Numerical Examples 399
3 Shell Stability Analysis with Implicit Dynamics 408
3.1 Shell Dynamic Formulation 408
3.2 Time-Stepping Schemes 410
3.3 Numerical Examples 411
4 Conclusions 418
References 419
Development of Invariant-Based Triangular Element for Nonlinear Thermoelastic Analysis of Laminated Shells 422
1 Introduction 422
2 Coordinate Transformation of Tensor Components 425
3 Template Formulas for Invariants 427
4 Invariant Representation of the Strain Energy Density of Anisotropic Shell 428
4.1 Representation of V1 in Terms of Combined Invariants 428
4.2 Representation of V2 in Terms of Combined Invariants 430
5 Potential Energy of the Shell as a Function of Natural Strains 431
6 Potential Energy of the Shell Finite Element 432
7 Solution Procedure 435
8 Numerical Results and Discussion 438
8.1 Thermal Postbuckling Behavior of a Cross-Ply Plate 439
8.2 Thermal Buckling and Postbuckling of Circular Cylindrical Shells 439
8.3 Thermal Buckling and Postbuckling of Conical Shells 443
8.4 Large Thermal Deflections of a Laminated Cylindrical Panel 446
9 Conclusions 449
References 449
Interaction of a Spherical Wave with a Rectangular Plate in a Ground 452
1 Introduction 452
2 Problem Statement of Interaction of Harmonic Spherical Wave Propagating in Continuum and a Three-Layered Plate 453
3 Dynamic Equations of Ground 454
4 Fourier Decomposition of Unknown Functions for both Media 456
5 Ingoing Harmonic Spherical Wave 457
6 The Plate Geometry 459
7 Fourier Decomposition of Unknown Plates' Functions 462
8 Conditions on the Contact Surface 463
9 Computing of the Fourier Coefficients for the Potentials in Ambient Media 463
References 465
Localized Parametric Vibrations of Laminated Cylindrical Shell Under Non-uniform Axial Load Periodically Varying with Time 467
1 Introduction 467
2 Non-linear Equations 469
3 Linearisation of Governing Equations and Additional Assumptions 473
4 Reduction of Equations to Dimensionless Form 474
5 Asymptotic Approach 475
6 Parametric Instability Domains 481
7 Examples 482
8 Conclusions 484
References 485
Numerical Analysis of Free Vibration of Laminated Thin-Walled Closed-Section Shell Structures 487
1 Introduction 487
2 Numerical Analysis 488
2.1 Box Shaped Shell 488
2.2 Tube Shaped Shell 493
3 Conclusions 497
References 500
Abnormal Buckling of Thin-Walled Bodies with Shape Memory Effects Under Thermally Induced Phase Transitions 501
1 Introduction 501
2 Equilibrium Stability Problem's Statement 505
2.1 General Statement of the Problem 505
3 Constitutive Equations for the Shape Memory Alloy Undergoing Thermoelastic Phase Transitions 507
3.1 Basics of the Coupled Phenomonological Model of Non-isothermal SMA Behavior 507
3.2 Incremental Representation of the Constitutive Equations 509
4 Stability Concepts in the Linearized Buckling Theory for thin-walled SMA structures 511
4.1 Extended von Kármán's Buckling Concept for SMA Structures 512
4.2 Extended Shenley's Concept for SMA Structures 512
5 Numerical Simulation of the Buckling and Postbuckling for Thin-Walled SMA Systems 513
5.1 Buckling Problem's Statement for a Prismatic SMA Beam 513
5.2 Solid Finite Element Model and Initial Camber Calibration 515
5.3 Simulation of the Buckling and Postbuckling of SMA Beams Under Direct Martensite Transitions 518
6 Conclusions 525
References 526
On the Homogenization of Nonlinear Shell 533
1 Introduction 533
1.1 Notation 537
1.2 Description of the Shell Geometry 538
1.3 The Bidimensional Macroscopic Nonlinear Shell Model 539
1.4 The Transition Law Between Macroscopic and Microscopic Scales and the Microscopic Problem 542
2 Conclusion 544
References 544
A Non-linear Theory of Thin-Walled Rods of Open Profile Deduced with Incremental Shell Equations 548
1 Introduction 548
2 Incremental Theory of Rods of Open Profile 549
2.1 Beam Kinematic Description 550
2.2 Governing Equations of the Incremental Beam Theory 552
3 Shell as a Material Surface 556
3.1 Differential Geometry of a Surface 556
3.2 Kinematics of Deformation 557
3.3 Equations 558
3.4 Boundary Conditions 559
3.5 Transformation to the Reference Configuration with Piola Tensors 560
3.6 Linearised Equations of a Pre-stressed Shell 561
4 Torsion of a Pre-stressed Cylindrical Panel 563
4.1 Pre-stressed State 563
4.2 Approximate Variational Approach 566
4.3 Exact Solution of the Shell Problem of Torsion 567
4.4 Torsion of a Pre-stressed Shell 568
5 Numerical Experiments and Validation 571
5.1 Linear Deformation Analysis 573
5.2 Stability Analysis: Critical Forces and Post-buckling Behaviour 577
6 Conclusions 580
References 581
Buckling of Elastic Circular Plate with Surface Stresses 583
1 Introduction 583
2 Governing Equations 584
3 Circular Plate with Surface Stresses 585
4 Equations of Neutral Equilibrium 586
5 Axisymmetric Buckling 590
6 Numerical Results 592
7 Conclusion 594
References 595
Asymptotic Derivation of Nonlinear Plate Models from Three-Dimensional Elasticity Theory 597
1 Introduction 597
2 Small-Thickness Estimate of the Energy 601
3 Membrane Limit 606
4 Pure Bending 609
5 Asymptotic Model for Combined Bending and Stretching 611
6 Reflection Symmetry and Ill-Posedness 615
7 Koiter's Model 616
8 Proposed Model 617
References 619
Selected Stability Problems of Thin-Walled Columns and Beams 621
1 Introduction 621
2 Global Buckling of Columns Geometrically Perfect and Imperfect 624
3 Local Buckling of Columns and Beams Made of Linearly and Non-linearly Elastic Materials 625
4 Global and Local Post-buckling of Columns and Beams Made of Linear and Non-linear Elastic Materials 627
5 Conclusions 628
References 628
Higher-Order Weak Formulation for Arbitrarily Shaped Doubly-Curved Shells 632
1 Introduction 632
2 Definition of the Geometry 634
2.1 Numerical Remarks 636
3 Shell Structural Model 639
3.1 Displacement Interpolation Using Lagrange Polynomials 640
3.2 Kinematic Equations 642
3.3 Constitutive Equations 643
3.4 Equations of Motion 645
4 Numerical Technique 648
4.1 Discrete Form of the Governing Equations 649
5 Free Vibration Analysis 650
6 Applications 651
6.1 Convergence Features 651
6.2 Arbitrarily Shaped Domains 653
7 Final Remarks 657
References 658
Strong Formulation: A Powerful Way for Solving Doubly Curved Shell Structures 663
1 Introduction 663
2 Principles of Differential Geometry 665
3 Shell Fundamental Equations 667
4 Numerical Procedure 674
5 Applications and Results 678
6 Final Remarks 684
References 684
On a Simple Shell Model for Thin Structures with Functionally Graded Materials 690
1 Introduction 690
2 Variational Formulation 692
2.1 Kinematics 692
2.2 Constitutive Equations for the Stress Resultants 693
2.3 Equilibrium Equations and a Constraint 696
2.4 Constrained Optimization Problem 697
2.5 Representative Volume Element 698
3 Examples 702
3.1 Simply Supported Square Plate Under Uniform Load 702
3.2 Simply Supported Square Sandwich Plate Under Uniform Load 708
3.3 Clamped Circular Shell 712
4 Conclusions 712
References 712
On Performance of Nine-Node Quadrilateral Shell Elements 9-EAS11 and MITC9i 714
1 Introduction 714
2 Basic Shell Equations with Drilling Rotation 715
3 Corrected Shape Functions for Nine-Node Shell Element 717
4 Characteristics of 9-EAS11 Shell Elements 718
5 Numerical Examples 722
5.1 Patch Tests 722
5.2 Curved Cantilever 723
5.3 Pinched Hemispherical Shell with Hole 726
6 Final Remarks 727
References 727
Higher Order Theory of Electro-Magneto-Elastic Plates and Shells 729
1 Introduction 729
2 3-D Electro-Magneto-Elasticity in Orthogonal Coordinates 732
3 3-D Electro-Magneto-Elasticity in Coordinates Related to the Middle Surface 737
4 2-D Formulation of the Problem 740
5 Higher Order Theory of Electro-Magneto-Elastic Plates in Cartesian Coordinates 746
6 Higher Order Theory of Electro-Magneto-Elastic Plates in Polar Coordinates 750
7 Higher Order Theory of Electro-Magneto-Elastic Cylindrical Shell 756
8 Higher Order Theory of Electro-Magneto-Elastic Spherical Shell 762
9 Conclusions 767
References 768
Exact Solutions of Nonlinear Micropolar Elastic Theory for Compressible Solids 772
1 Introduction 772
2 Initial Relations of Nonlinear Micropolar Elasticity 773
3 Cylindrical Bending of Rectangular Plate 777
3.1 Subfamily 1A 777
3.2 Subfamily 1B 778
3.3 Subfamily 1C 779
3.4 Subfamily 1D 780
4 Straightening of a Circular Hollow–Cylinder Sector 781
4.1 Subfamily 2A 781
4.2 Subfamily 2B 783
4.3 Subfamily 2C 783
4.4 Subfamily 2D 784
5 Pure Bending and Reversing of a Circular Hollow–Cylinder Sector 785
5.1 Subfamily 3A 786
5.2 Subfamily 3B 787
5.3 Subfamily 3C 788
5.4 Subfamily 3D 788
6 Double Cylindrical Bending of a Circular Hollow–Cylinder Sector 789
6.1 Subfamily 4A 790
6.2 Subfamily 4B 791
6.3 Subfamily 4C 792
6.4 Subfamily 4D 792
7 Radially Symmetric Deformation of a Hollow Sphere 793
7.1 Subfamily 5A 794
7.2 Subfamily 5B 795
8 Reversing of a Hollow Sphere 796
8.1 Subfamily 6A 796
8.2 Subfamily 6B 797
9 Conclusion 797
References 798
| Erscheint lt. Verlag | 25.9.2019 |
|---|---|
| Reihe/Serie | Advanced Structured Materials | Advanced Structured Materials |
| Zusatzinfo | XXII, 798 p. 276 illus., 139 illus. in color. |
| Sprache | englisch |
| Themenwelt | Naturwissenschaften ► Physik / Astronomie |
| Technik ► Maschinenbau | |
| Schlagworte | Dimension-reduced Structure Mechanics Models • Material models • Pietraszkiewicz • Plates and Shells • Thin Structures • timoshenko |
| ISBN-10 | 3-030-17747-5 / 3030177475 |
| ISBN-13 | 978-3-030-17747-8 / 9783030177478 |
| Informationen gemäß Produktsicherheitsverordnung (GPSR) | |
| Haben Sie eine Frage zum Produkt? |
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