Rotation, reflection, and frame changes : orthogonal tensors in computational engineering mechanics /

• Κύριος συγγραφέας: Brannon, R. M. (Rebecca M.) (συγγραφέας.)
• Μορφή: Ηλ. βιβλίο
• Γλώσσα: English
• Έκδοση: Bristol : IOP Publishing, c2018.
• Σειρά: IOP expanding physics
• Θέματα: Μαθηματικά για μηχανικούς; Μηχανική, Εφαρμοσμένη; Μηχανική, Αναλυτική

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...

Πίνακας περιεχομένων:

• 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.