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source: openmichigan 2014年5月30日

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The idea for these Lectures on Continuum Physics grew out of a short series of talks on materials physics at University of Michigan, in the summer of 2013. Those talks were aimed at advanced graduate students, post-doctoral scholars, and faculty colleagues. From this group the suggestion emerged that a somewhat complete set of lectures on continuum aspects of materials physics would be useful. The lectures that you are about to dive into were recorded over a six-week period at the University. Given their origin, they are meant to be early steps on a path of research in continuum physics for the entrant to this area, and I daresay a second opinion for the more seasoned exponent of the science. The potential use of this series as an enabler of more widespread research in continuum physics is as compelling a motivation for me to record and offer it, as is its potential as an open online class.

This first edition of the lectures appears as a collection of around 130 segments (I confess, I have estimated, but not counted) of between 12 and 30 minutes each. The recommended single dose of online instruction is around 15 minutes. This is a recommendation that I have flouted with impunity, hiding behind the need to tell a detailed and coherent story in each segment. Still, I have been convinced to split a number of the originally longer segments. This is the explanation for the proliferation of Parts I, II and sometimes even III, with the same title. Sprinkled among the lecture segments are responses to questions that arose from a small audience of students and post-doctoral scholars who followed the recordings live. There also are assignments and tests.

The roughly 130 segments have been organized into 13 units, each of which may be a chapter in a book. The first 10 units are standard fare from the continuum mechanics courses I have taught at University of Michigan over the last 14 years. As is my preference, I have placed equal emphasis on solids and fluids, insisting that one cannot fully appreciate the mechanical state of one of these forms of matter without an equal appreciation of the other. At my pace of classroom teaching, this stretch of the subject would take me in the neighborhood of 25 lectures of 80 minutes each. At the end of the tenth of these units, I have attempted, perhaps clumsily, to draw a line by offering a roadmap of what the viewer could hope to do with what she would have learned up to that point. It is there that I acknowledge the modern masters of continuum mechanics by listing the books that, to paraphrase Abraham Lincoln, will enlighten the reader far above my poor power to add or detract.

At this point the proceedings also depart from the script of continuum mechanics, and become qualified for the mantle of Continuum Physics. The next three units are on thermomechanics, variational principles and mass transport--subjects that I have learned from working in these areas, and have been unable to incorporate in regular classes for a sheer want of time. In the months and years to come, new editions of these Lectures on Continuum Physics will feature an enhancement of breadth and depth of these three topics, as well as topics in addition to them.

Finally, a word on the treatment of the subject: it is mathematical. I know of no other way to do continuum physics. While being rigorous (I hope) it is, however, neither abstract nor formal. In every segment I have taken pains to make connections with the physics of the subject. Props, simple but instructive, have been used throughout. A deformable plastic bottle, water and food color have been used--effectively, I trust. The makers of Lego, I believe, will find reason to be pleased. Finally, the time-honored continuum potato has been supplanted by an icon of American life: the continuum football.

Krishna Garikipati

Ann Arbor, December 2013

01.01. Introduction 18:20

01.01. Response to a question 1:34

01.02. Vectors I 14:56

01.02. Response to a question 2:04

01.03. Vectors II 27:25

01.04. Vectors III 24:41

02.01. Tensors I 15:31

02.02. Tensors II 11:59

02.02. Response to a question 17:18

02.03. Tensors III 21:16

02.04. Tensor properties I 14:04

02.05. Tensor properties I 16:07

02.06. Tensor properties II 15:43

02.07. Tensor properties II 13:55

02.08. Tensor properties III 1:02

02.09. Vector and tensor fields 9:22

02.10. Vector and tensor fields 16:54

03.01. Configurations 14:29

03.02. Configurations 14:31

03.03. Motion 18:32

03.03. Response to a question 2:56

03.03. Response to a follow up question 2:59

03.04. The Lagrangian description of motion 14:32

03.05. The Lagrangian description of motion 15:35

03.06. The Eulerian description of motion 14:10

03.07. The Eulerian description of motion 14:07

03.08. The material time derivative 14:11

03.09. The material time derivative 16:52

03.09. Response to a question 1:21

04.01. The deformation gradient: mapping of curves 23:41

04.02. The deformation gradient: mapping of surfaces and volumes 18:08

04.03. The deformation gradient: mapping of surfaces and volumes 14:26

04.04. The deformation gradient: a first order approximation of the deformation 22:51

04.05. Stretch and strain tensors 21:00

04.06. Stretch and strain tensors 10:14

04.06. Response to a question 4:43

04.07. The polar decomposition I 22:03

04.08. The polar decomposition I 11:31

04.09. The polar decomposition II 14:21

04.10. The polar decomposition II 16:12

04.10. Response to a question 3:00

04.11. Velocity gradients, and rates of deformation 15:36

04.12. Velocity gradients, and rates of deformation 15:46

05.01. Balance of mass I 19:35

05.02. Balance of mass I 8:33

05.03. Balance of mass II 27:08

05.04. Balance of mass II 14:51

05.05. Reynolds' transport theorem I 19:07

05.06. Reynolds' transport theorem I 10:25

05.07. Reynolds' transport theorem II 20:36

05.08. Reynolds' transport theorem III 23:33

05.08. Response to a question 8:47

05.09. Linear and angular momentum I 24:38

05.10. Linear and angular momentum II 18:18

05.11. The moment of inertia tensor 16:41

05.12. The moment of inertia tensor 27:50

05.13. The rate of change of angular momentum 18:37

05.14. The balance of linear and angular momentum for deformable, continuum bodies 26:31

05.15. The balance of linear and angular momentum for deformable, continuum bodies 18:38

05.16. The Cauchy stress tensor 26:17

05.17. Stress-- An Introduction 19:55

06.01. Balance of energy 23:03

06.01. Response to a question 11:13

06.01. Response to a follow up question 6:58

06.02. Additional measures of stress 25:05

06.03. Additional measures of stress 15:54

06.03. Response to a question 6:50

06.03. Response to a follow up question 12:34

06.04. Work conjugate forms 25:57

06.05. Balance of linear momentum in the reference configuration 29:06

07.01. Equations and unknowns--constitutive relations 15:29

07.01. Response to a question 5:35

07.02. Constitutitve equations 27:03

07.03. Elastic solids and fluids--hyperelastic solids 29:16

07.03. Response to a question 5:54

08.01. Objectivity--change of observer 17:44

08.02. Objectivity--change of observer 14:47

08.03. Objective tensors, and objective constitutive relations 20:10

08.04. Objective tensors, and objective constitutive relations 11:06

08.05. Objectivity of hyperelastic strain energy density functions 29:18

08.06. Examples of hyperelastic strain energy density functions 23:21

08.07. Examples of hyperelastic strain energy density functions 10:08

08.07. Response to a question 4:36

08.08. The elasticity tensor in the reference configuration 14:27

08.09. Elasticity tensor in the current configuration--objective rates 25:00

08.10. Elasticity tensor in the current configuration--objective rates 11:40

08.11. Objectivity of constitutive relations for viscous fluids 19:06

08.12. Models of viscous fluids 28:15

08.12. Response to a question 3:50

08.13. Summary of initial and boundary value problems of continuum mechanics 25:21

08.14. An initial and boundary value problem of fluid mechanics--the Navier Stokes equations 18:23

08.15. An initial and boundary value problem of fluid mechanics--the Navier Stokes equation 14:58

08.16. An initial and boundary value problem of fluid mechanics II 20:38

08.17. Material symmetry 1--Isotropy 28:26

08.17. Response to a question 3:18

08.18. Material symmetry 2--Isotropy 11:04

08.19. Material symmetry 2--Isotropy 23:16

08.20. Material symmetry 3--Isotropy 23:54

09.01. A boundary value problem in nonlinear elasticity I 17:16

09.02. A boundary value problem in nonlinear elasticity I 13:25

09.02. Response to a question 7:15

09.03. A boundary value problem in nonlinear elasticity II--The inverse method 17:43

09.03. Response to another question 12:02

10.01. Linearized elasticity I 12:19

10.02. Linearized elasticity I 18:36

10.03. Linearized elasticity II 16:03

10.04. Linearized elasticity II 16:24

10.04. Response to a question 3:44

10.05. Classical continuum mechanics: Books, and the road ahead 22:26

11.01. The first law of thermodynamics the balance of energy 15:57

11.02. The first law of thermodynamics the balance of energy 12:30

11.03. The first law of thermodynamics the balance of energy 16:41

11.04. The second law of thermodynamics the entropy inequality 14:48

11.05. Legendre transforms the Helmholtz potential 20:23

11.06. The Clausius Planck inequality 17:24

11.07. The Clausius Duhem inequality 22:57

11.07. Response to a question 5:33

11.08. The heat transport equation 19:11

11.09. Thermoelasticity 25:23

11.10. The heat flux vector in the reference configuration 22:53

12.01. The free energy functional 12:37

12.02. The free energy functional 19:59

12.03. Extremization of the free energy functional variational derivatives 26:44

12.04. Euler Lagrange equations corresponding to the free energy functional 27:36

12.05. The weak form and strong form of nonlinear elasticity 23:42

12.06. The weak form and strong form of nonlinear elasticity 20:17

13.01. The setting for mass transport 19:20

13.02. The setting for mass transport 11:38

13.03. Aside A unified treatment of boundary conditions 20:25

13.04. The chemical potential 20:19

13.05. The chemical potential 14:52

13.06. Phase separation non convex free energy 16:05

13.07. Phase separation non convex free energy 17:31

13.08. The role of interfacial free energy 27:07

13.09. The Cahn Hilliard formulation 23:27

13.10. The Cahn Hilliard formulation 18:22

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