Rotation, reflection, and frame changes : orthogonal tensors in computational engineering mechanics / R.M. Brannon.

Format:

Book

Author/Creator:

Brannon, R. M. (Rebecca M.), author.

Contributor:

Institute of Physics (Great Britain), publisher.

Series:

IOP (Series). Release 4.

IOP expanding physics

[IOP release 4]

IOP expanding physics, 2053-2563

Language:

English

Subjects (All):

Engineering mathematics.

Mechanics, Applied.

Mechanics, Analytic.

Physical Description:

1 online resource (various pagings) : color illustrations.

Other Title:

Orthogonal tensors in computational engineering mechanics.

Place of Publication:

Bristol [England] : IOP Publishing, [2018]

System Details:

Mode of access: World Wide Web.

System requirements: Adobe Acrobat Reader, EPUB reader, or Kindle reader.

text file

Biography/History:

For over 25 years, first as a principal researcher (and manager) at Sandia National Laboratories and more recently as an Associate Professor of Mechanical Engineering at the University of Utah and ASME fellow, Dr Brannon has developed practical engineering constitutive models for brittle and ductile material failure at high strain rates and large strains. Her research has investigated a wide range of materials including piezoelectric ceramics, armor ceramics, geological materials, energetic materials, and metals (usually for high-rate applications). Constitutive models she has developed are used in DoD and DOE production codes such as CTH and ALEGRA. Applications have included protective structures, underground structure integrity, electroactive power supplies, artificial hip implant rapid materials ranking, shock-induced vaporization, in vivo measurements of callus strains, and other numerous other problems in the applied sciences. Dr Brannon is particularly known for her monographs on tensor analysis, plasticity, code portability, code verification, and massive deformation kinematics in the material point method.

Summary:

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-contained comprehensive review that covers both elementary and advanced concepts in a manner comprehensible to engineers is rare. Rotation, Reflection, and Frame Changes surveys a refreshingly broad range of rotation-related research that is routinely needed in engineering practice. By illustrating key concepts in computer source code, this book stands out as an unusually accessible guide for engineers and scientists in engineering mechanics.

Contents:

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.

Notes:

"Version: 20180401"--Title page verso.

Includes bibliographical references.

Title from PDF title page (viewed on May 4, 2018).

Other Format:

Print version:

ISBN:

9780750314541

9780750314534

OCLC:

1034808900

Access Restriction:

Restricted for use by site license.