This book presents the theory of plates and shells on the basis of the three-dimensional parent theory. The authors explore the thinness of the structure to represent the mechanics of the actual thin three-dimensional body under consideration by a more tractable two-dimensional theory associated with an interior surface. In this way, the relatively complex three-dimensional continuum mechanics of the thin body is replaced by a far more tractable two-dimensional theory. To ensure that the resulting model is predictive, it is necessary to compensate for this ‘dimension reduction’ by assigning additional kinematical and dynamical descriptors to the surface whose deformations are modelled by the simpler two-dimensional theory. The authors avoid the various ad hoc assumptions made in the historical development of the subject, most notably the classical Kirchhoff–Love hypothesis requiring that material lines initially normal to the shell surface remain so after deformation. Instead, such conditions, when appropriate, are here derived rather than postulated.
Author(s): David J. Steigmann, Mircea Bîrsan, Milad Shirani
Series: Solid Mechanics and Its Applications, 274
Publisher: Springer
Year: 2023
Language: English
Pages: 257
City: Cham
Preface
Contents
1 Tensor Analysis in Euclidean Space Using Curvilinear Coordinates
1.1 Cartesian and Curvilinear Coordinates
1.2 Representations of Vectors and Tensors
1.2.1 Vectors
1.2.2 Representation in Cartesian Coordinates
1.2.3 Tensors
1.2.4 Coordinate Transformations
1.3 Differential Operators in Curvilinear Coordinates
1.3.1 Gradients
1.3.2 Divergence
1.3.3 Product Rule for Covariant Derivatives
1.4 Orthogonal Coordinates
1.5 Exercises
References
2 Local Geometry of Deformation
2.1 Description of the Deformation
2.1.1 Deformation Tensor
2.1.2 Volumes and Areas
2.2 Displacement, Strain, and Stress
2.3 Local Geometry of Surfaces
2.4 Differentiation on Surfaces. Curvature
2.4.1 Curvature of a Curve on the Surface
2.4.2 Mean Curvature and Gaussian Curvature of the Surface
2.5 Compatibility Conditions for the Fundamental Tensors
2.6 Green-Stokes Formula
2.7 Surface Differential Operators
2.8 Exercises
References
3 Hyperelastic Solids: Purely Mechanical Theory
3.1 Constitutive Relations
3.2 Small Strains
3.2.1 Special Case: Isotropy
3.3 Potential Energy
3.4 Legendre-Hadamard Condition for Stability
References
4 Linearly Elastic Plates
4.1 Three-dimensional Theory
4.2 Derivation of the Plate Model
4.2.1 Integration over the Thickness
4.2.2 Edge Loads
4.2.3 Optimal Expressions
4.2.4 Lateral Loads
4.3 Equilibrium Equations
4.4 Isotropic Plates
4.4.1 Displacement Gradients
4.4.2 Strain Energy
4.4.3 Decoupled Equilibrium Equations
4.5 Energy Minimizers
4.6 Exercises
References
5 Linear Shell Theory
5.1 Geometry of the Curved Shell
5.2 Kinematics
5.3 Derivation of the Shell Model
5.4 Optimal Expression for Potential Energy
5.4.1 Membrane Energy
5.4.2 Pure Bending
5.4.3 Estimate of the Coupling Term in the Energy
5.5 Measures of Distortion
5.6 Isotropy
5.7 Equilibrium Equations and Boundary Conditions
5.8 Classical Membrane Theory
5.8.1 Equilibrium Under Normal Pressure
5.9 Bending Theory
5.9.1 Circular Cylindrical Shells
5.10 Exercises
References
6 Nonlinear Equations for Plates and Shells
6.1 Asymptotic Derivation of Nonlinear Plate Models
6.1.1 Background from Three-Dimensional Nonlinear Elasticity Theory
6.1.2 Small-Thickness Estimate of the Energy
6.1.3 Membrane Limit
6.1.4 Pure Bending
6.1.5 Asymptotic Model for Combined Bending and Stretching
6.1.6 Reflection Symmetry and Ill-Posedness
6.1.7 Koiter's Model
6.2 Koiter's Shell Theory
6.2.1 Geometric and Kinematic Formulae
6.2.2 Expansion of the Three-Dimensional Energy
6.2.3 The Optimum Order—h3 Energy
6.2.4 Koiter's Energy as the Leading-Order Model
6.2.5 The Explicit Energy for Isotropic Materials
6.3 Equilibrium Equations
6.4 Exercises
References
7 Buckling of Elastic Plates
7.1 Nonlinear Elasticity and Stability
7.2 Linear Incremental Elasticity and Bifurcation of Equilibria
7.3 Two–dimensional Model for Plate-Buckling
7.3.1 Description of Plates as Thin Prismatic Bodies
7.3.2 Thickness-Wise Power Expansion of the Second Variation
7.3.3 The Refined Model
7.3.4 Conditions Pertaining to Reflection Symmetry and the Pre-stress
7.3.5 Derivation of the Classical Model
7.4 Exercises
References
8 Saint-Venant Problem for General Cylindrical Shells
8.1 Relaxed Saint-Venant's Problem for Cylindrical Shells
8.2 Closed–form Exact Solutions
8.2.1 Extension-Bending-Torsion Problem
8.2.2 Flexure Problem
8.3 Simplified Solution
8.4 Circular Cylindrical Tubes
References
