This book presents the foundation and validation of the Cosserat Plate Theory, numerical experiments of deformation and vibration, and the unique properties of the Cosserat plates. Our approach incorporates the high accuracy assumptions of the Cosserat plate deformation consistent with the Cosserat Elasticity equilibrium equations, constitutive formulas, strain-displacement and torsion-microrotation relations. The Cosserat Plate Theory is parametric, where the "splitting parameter" minimizes the Cosserat plate energy. The validation of the theory is based on the comparison with the three-dimensional Cosserat Elastostatics and Elastodynamics. The numerical results are obtained using the Finite Element Method (FEM) specifically developed to solve the parametric system of equations. The analysis of deformation of a variety of Cosserat plates shows the stress concentration reduction, higher stiffness of Cosserat plates, and the size-effect related to the microstructure. The analysis of vibration of Cosserat plates predicts size-related properties of the plate vibration, the existence of the additional so-called Cosserat plate resonances, and the dynamic anisotropy, related to the dependency of the resonances on the microelement's shapes and orientations.
Author(s): Lev Steinberg, Roman Kvasov
Publisher: CRC Press
Year: 2022
Language: English
Pages: 182
City: Boca Raton
Cover
Half Title
Title Page
Copyright Page
Dedication
Contents
Preface
Acknowledgments
Introduction
Notations
CHAPTER 1: Cosserat Elasticity
1.1. SCOPE OF THE CHAPTER
1.2. COSSERAT ELASTICITY
1.3. LINEAR THEORY OF COSSERAT ELASTICITY
1.3.1. Cosserat Space
1.3.2. Cosserat Body Deformation
1.3.3. Infinitesimal Cosserat Deformation and Motion
1.3.4. Conservation of Mass and Microinertia
1.3.5. Conservation of Linear and Angular Momentum
1.3.6. Law of Conservation of Energy
1.3.7. Cosserat Elastic Energy
1.4. VARIATIONAL PRINCIPLE FOR COSSERAT ELASTOSTATICS
1.5. PRINCIPLE OF MINIMUM POTENTIAL ENERGY
1.6. VARIATIONAL PRINCIPLE FOR COSSERAT ELASTODYNAMICS
CHAPTER 2: Analysis of Cosserat Sample Bodies
2.1. SCOPE OF THE CHAPTER
2.2. COSSERAT SAMPLE BODIES
2.3. STATICS OF COSSERAT SAMPLE BODIES
2.4. DYNAMICS OF COSSERAT SAMPLE BODIES
2.5. PLATE GEOMETRY RESTRICTION FOR COSSERAT BODIES
CHAPTER 3: Cosserat Plate Statics
3.1. SCOPE OF THE CHAPTER
3.2. PLATE ASSUMPTIONS
3.2.1. Boundary Conditions
3.2.2. Stress Approximations
3.2.3. Couple Stress Approximations
3.2.4. Kinematic Approximations
3.3 COSSERAT PLATE STATIC EQUILIBRIUM EQUATIONS
3.4. STRESS ENERGY OF THE COSSERAT PLATE
3.5. STRAIN-DISPLACEMENT AND TORSION-MICROROTATION RELATIONS
3.6. CONSTITUTIVE FORMULAS FOR COSSERAT PLATE
3.7. VARIATIONAL FORMULATION OF COSSERAT PLATE BENDING
3.8. OPTIMAL VALUE OF THE SPLITTING PARAMETER
3.9. STATICS FIELD EQUATIONS
CHAPTER 4: Cosserat Plate Dynamics
4.1. SCOPE OF THE CHAPTER
4.2. PLATE ASSUMPTIONS
4.2.1. Boundary and Initial Conditions
4.2.2. Stresses and Kinematic Approximations
4.3. DYNAMIC FIELD EQUATIONS
CHAPTER 5: Validation of the Cosserat Plate Theory
5.1. SCOPE OF THE CHAPTER
5.2. COSSERAT SAMPLE PLATE
5.3. VALIDATION OF THE COSSERAT PLATE STATICS
5.4. VALIDATION OF THE COSSERAT PLATE DYNAMICS
CHAPTER 6: Finite Element Method for Cosserat Plates
6.1. SCOPE OF THE CHAPTER
6.2. CLAMPED COSSERAT PLATE BENDING PROBLEM
6.3. FINITE ELEMENT ALGORITHM FOR COSSERAT ELASTIC PLATES
6.4. SOLUTIONS SPACE FOR THE CLAMPED COSSERAT PLATE
6.5. WEAK FORMULATION OF THE COSSERAT PLATE BENDING PROBLEM
6.6. CONSTRUCTION OF THE FINITE ELEMENT SPACES
6.7. CALCULATION OF THE STIFFNESS MATRIX AND THE LOAD VECTOR
6.8. NUMERICAL RESULTS
CHAPTER 7: Validation of the Finite Element Computation
7.1. SCOPE OF THE CHAPTER
7.2. SAMPLE COSSERAT PLATE
7.3. VALIDATION OF THE FEM FOR DIRICHLET BOUNDARY CONDITIONS
7.4. VALIDATION OF THE FEM FOR MIXED DIRICHLET-NEUMANN BOUNDARY CONDITIONS
7.5. VALIDATION OF THE FEM FOR SIMPLY SUPPORTED COSSERAT PLATE
CHAPTER 8: Unique Properties of Cosserat Plates
8.1. SCOPE OF THE CHAPTER
8.2. COSSERAT SAMPLE PLATE
8.3. COSSERAT PLATE STATICS
8.3.1. Plate Stiffness
8.3.2. Stress Concentration
8.4. COSSERAT PLATE DYNAMICS
8.4.1. Plate Resonances
8.4.2. Dynamic Anisotropy
Appendix A
Appendix B
Appendix C
Bibliography
Index
