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Rotation, reflection, and frame changes : orthogonal tensors in computational engineering mechanics /
Whilst vast literature is available for the most common rotation-related tasks such as coordinate changes, most reference books tend to cover one or two methods, and resources for less-common tasks are scarce. Specialized research applications can be found in disparate journal articles, but a self-c...
| Clasificación: | Libro Electrónico |
|---|---|
| Autor principal: | |
| Formato: | Electrónico eBook |
| Idioma: | Inglés |
| Publicado: |
Bristol [England] (Temple Circus, Temple Way, Bristol BS1 6HG, UK) :
IOP Publishing,
[2018]
|
| Colección: | IOP (Series). Release 4.
IOP expanding physics. |
| Temas: | |
| Acceso en línea: | Texto completo |
Tabla de Contenidos:
- 1. Introduction
- 2. Notation and tensor analysis essentials
- 2.1. Linear fractional transform
- 2.2. Visualizing rotations
- 3. Orthogonal basis and coordinate transformations
- 3.1. Superimposed rotations
- 3.2. Basis rotations
- 4. Rotation operations
- 4.1. Why apparent inconsistency in placement of the negative sign?
- 5. Axis and angle of rotation
- 5.1. Euler-Rodrigues formula
- 5.2. Computing the rotation tensor given axis and angle
- 5.3. Corollary to the Euler-Rodrigues formula : existence of a preferred basis
- 5.4. Computing axis and angle given the rotation tensor
- 6. Rotations contrasted with reflections
- 7. Quaternion representation of a rotation
- 7.1. Shoemake's form
- 7.2. Relationship between quaternion and axis/angle forms
- 8. Dyadic form of an invertible linear operator
- 8.1. Special case : lab basis
- 8.2. Special case : dyadic form of a rotation operation
- 8.3. Constructing a rotation that will transform one specified vector to another specified vector
- 8.4. Constructing a rotation from knowledge of initial and final 'marker' locations in a body
- 9. Sequential rotations
- 9.1. The distinction between fixed and follower axes
- 9.2. Roll, pitch, yaw : sequential rotations about fixed (laboratory) axes
- 9.3. Euler angles : sequential rotations about 'follower' axes
- 10. Series expression for a rotation
- 10.1. Cayley transformations
- 11. Spectrum of a rotation
- 12. Polar decomposition
- 12.1. Difficult definition of the deformation gradient
- 12.2. Intuitive definition of the deformation gradient
- 12.3. The Jacobian of the deformation
- 12.4. Invertibility of a deformation
- 12.5. Sequential deformations
- 12.6. Matrix analysis version of the polar-decomposition theorem
- 12.7. Polar decomposition--a hindsight intuitive interpretation
- 12.8. Variational interpretation of the polar decomposition
- 12.9. A more rigorous (classical) presentation of the polar-decomposition theorem
- 12.10. The 'fast' way to do a polar decomposition in two dimensions
- 12.11. Scaling properties of a polar decomposition
- 12.12. Classic method for obtaining a polar decomposition in 3D
- 12.13. Another iterative polar decomposition in 3D
- 13. Strain measures
- 13.1. One-dimensional strain measures
- 13.2. Three-dimensional strain definitions
- 14. Remapping, advecting, or interpolating rotations
- 14.1. Proposal 1 : Map and re-compute the polar decomposition
- 14.2. Proposal 2 : Discard the 'stretch' part of a mixed rotation
- 14.3. Proposal 3 : Advect the pseudo-rotation vectors
- 14.4. Proposal 4 : Mix the quaternions
- 14.5. Advection enhancement strategy #1 : solve the compatibility equations
- 14.6. Mixing enhancement strategy #2 : Lagrangian tracers
- 15. Rates and other derivatives of rotation
- 15.1. The 'spin' tensor
- 15.2. The angular velocity vector
- 15.3. Angular velocity in terms of axis and angle of rotation
- 15.4. Derivatives of rotation with respect to angle and axis
- 15.5. Difference between vorticity and polar spin
- 15.6. The (commonly mis-stated) Gosiewski's theorem
- 15.7. Rates of sequential rotations
- 15.8. Rates of simultaneous rotations
- 15.9. Integration of rotation rates
- 16. Variations of tensor-valued functions of scalars and vectors
- 16.1. A motivational example
- 16.2. A comment about rates of proper functions
- 16.3. The time rate of a principal function of a symmetric tensor
- 16.4. Time rate of the logarithmic strain
- 17. Statistics of random orientation
- 17.1. Elementary probability and statistics refresher
- 17.2. Uniformly random unit vectors--the theory
- 17.3. Uniformly random unit vectors--alternative implementation
- 17.4. 'Centroidally random' unit vectors
- 17.5. 'Nautical' visualization of a rotation
- 17.6. Uniformly random rotations
- 17.7. A basic algorithm for generating a uniformly random rotation
- 17.8. Generalization to generate transversely isotropic orientation distributions
- 17.9. Alternative algorithm for generating a uniformly random rotation
- 17.10. Shoemake's algorithm for uniformly random rotations
- 18. Introduction to material and tensor symmetries
- 18.1. Anisotropy classification via group theory
- 18.2. Quantifying and visualizing orientations
- 19. Frame indifference
- 19.1. A 3D spring--who expected it would be this hard!?
- 19.2. Introduction to frame indifference
- 19.3. Kinematics changes under superimposed rigid motion
- 19.4. Mechanics principles frame change
- 20. Tensor symmetry (not material symmetry)
- 20.1. What is isotropy of a tensor?
- 20.2. Isotropic second-order tensors in 3D space
- 20.3. Isotropic second-order tensors in 2D space
- 20.4. Isotropic fourth-order tensors in 3D
- 20.5. The isotropic part of a fourth-order tensor
- 20.6. Tensor transverse isotropy
- 20.7. Material transverse isotropy
- 21. Scalars and invariants
- 22. PMFI for incremental constitutive models
- 22.1. A frame-indifferent spring rate equation
- 22.2. The PMFI in general
- 22.3. PMFI in rate forms of the constitutive equations
- 22.4. Co-rotational rates (convected, Jaumann, polar, etc)
- 22.5. Lie derivatives and reference configurations
- 22.6. Frame indifference is only an essential (not final) step
- 23. Rigid-body mechanics
- 23.1. Rate of rotation
- 23.2. The slope-intercept of rigid motion
- 23.3. The point-slope description of rigid motion
- 23.4. Velocity and angular velocity for rigid motion
- 23.5. Time rate of a vector embedded in a rigid body
- 23.6. Acceleration for rigid motion
- 23.7. Important properties of a rigid body
- 23.8. Linear momentum of a rigid body
- 23.9. Angular momentum of a rigid body
- 23.10. Kinetic energy of a rigid body
- 23.11. Newton's equation (balance of linear momentum)
- 23.12. Euler's equation (balance of angular momentum)
- 24. Pseudo-body force for spinning problems
- 24.1. Kinematics of superimposed rotation (general analysis)
- 24.2. Fiducial body force for superimposed rigid motion
- 25. Computer graphics visualization
- 25.1. Orientation of the body
- 25.2. Mapping from the body to the screen
- 25.3. Mapping from the screen to the virtual visible surface
- 25.4. Changing the screen image of a body
- 26. Voigt and Mandel components
- 26.1. An introductory 3D example
- 26.2. Voigt components (inefficient and error prone!)
- 26.3. Mandel components (nice!)
- 26.4. Voigt components of fourth-order minor-symmetric tensors
- 26.5. Mandel components of fourth-order minor-symmetric tensors
- 26.6. Mandel components of fourth-order general tensors
- 26.7. Fourth-order linear transformations
- 26.8. Spectral analysis of fourth-order tensors
- 27. Higher-order rotations
- 27.1. Rotators : fourth-order rotations in Mandel form
- 27.2. Fourth-order 'focused identity' (projection) tensors
- 27.3. Focused rotations
- 27.4. Components of focused identities and elided projectors
- 27.5. Single-plane fourth-order rotations
- 27.6. Preferred basis for single-plane rotation
- 27.7. Double-plane fourth-order rotations
- 27.8. Multi-plane fourth-order rotations
- 28. Closing remarks.