Welcome to Lectures on Continuum Physics
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, postdoctoral 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 sixweek 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 postdoctoral 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 transportsubjects 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 usedeffectively, I trust. The makers of Lego, I believe, will find reason to be pleased. Finally, the timehonored continuum potato has been supplanted by an icon of American life: the continuum football.
Krishna Garikipati
Ann Arbor, December 2013
About the Creators
Dr. Garikipati's work draws from nonlinear mechanics, materials physics, applied mathematics and numerical methods. He's particularly interested in problems of mathematical biology, biophysics and the materials physics. Current research interests include: (1) mathematical and physical modelling of tumor growth, (2) cell mechanics (3) chemomechanically driven phenomena in materials, such as phase transformations and stressinfluenced mass transport. more...
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Final Homework 
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Midterm 1 
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Midterm 2 
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Midterm 3 
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Problem Set 1 
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Problem Set 2 
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Problem Set 3 
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Problem Set 4 
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Problem Set 5 
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Problem Set 6 
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01.01. Introduction 
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01.01. Response to a question 
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01.02. Response to a question 
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01.02. Vectors I 
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01.03. Vectors II 
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01.04. Vectors III 
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02.01. Tensors I 
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02.02. Response to a question 
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02.02. Tensors II 
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02.03. Tensors III 
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02.04. Tensor properties I 
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02.05. Tensor properties I 
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02.06. Tensor properties II 
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02.07. Tensor properties II 
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02.08. Tensor properties III 
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02.09. Vector and tensor fields 
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02.10. Vector and tensor fields 
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03.01. Configurations 
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03.02. Configurations 
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03.03. Motion 
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03.03. Response to a followup question 
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03.03. Response to a question 
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03.04. The Lagrangian description of motion 
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03.05. The Lagrangian description of motion 
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03.06. The Eulerian description of motion 
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03.07. The Eulerian description of motion 
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03.08. The material time derivative 
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03.09. Response to a question 
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03.09. The material time derivative 
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04.01. The deformation gradient: mapping of curves 
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04.02. The deformation gradient: mapping of surfaces and volumes 
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04.03. The deformation gradient: mapping of surfaces and volumes 
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04.04. The deformation gradient: a firstorder approximation of the deformation 
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04.05. Stretch and strain tensors 
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04.06. Response to a question 
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04.06. Stretch and strain tensors 
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04.07. The polar decomposition I 
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04.08. The polar decomposition I 
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04.09. The polar decomposition II 
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04.10. Response to a question 
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04.10. The polar decomposition II 
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04.11. Velocity gradients, and rates of deformation 
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04.12. Velocity gradients, and rates of deformation 
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05.01. Balance of mass I 
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05.02. Balance of mass I 
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05.03. Balance of mass II 
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05.04. Balance of mass II 
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05.05. Reynolds' transport theorem I 
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05.06. Reynolds' transport theorem I 
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05.07. Reynolds' transport theorem II 
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05.08. Response to a question 
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05.08. Reynolds' transport theorem III 
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05.09. Linear and angular momentum I 
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05.09. Response to a question 
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05.10. Linear and angular momentum II 
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05.11. The moment of inertia tensor 
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05.12. The moment of inertia tensor 
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05.13. The rate of change of angular momentum 
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05.14. The balance of linear and angular momentum for deformable, continuum bodies 
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05.15. The balance of linear and angular momentum for deformable, continuum bodies 
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05.16. The Cauchy stress tensor 
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05.17. Stress  An Introduction 
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06.01. Balance of energy 
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06.01. Response to a followup question 
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06.01. Response to a question 
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06.02. Additional measures of stress 
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06.03. Additional measures of stress 
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06.03. Response to a followup question 
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06.03. Response to a question 
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06.04. Work conjugate forms 
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06.05. Balance of linear momentum in the reference configuration 
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07.01. Equations and unknowns  constitutive relations 
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07.01. Response to a question 
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07.02. Constitutitve equations 
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07.03. Elastic solids and fluids  hyperelastic solids 
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07.03. Response to a question 
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08.01. Objectivity  change of observer 
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08.02. Objectivity  change of observer 
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08.03. Objective tensors, and objective constitutive relations 
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08.04. Objective tensors, and objective constitutive relations 
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08.05. Objectivity of hyperelastic strain energy density functions 
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08.06. Examples of hyperelastic strain energy density functions 
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08.07. Examples of hyperelastic strain energy density functions 
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08.07. Response to a question 
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08.08. The elasticity tensor in the reference configuration 
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08.09. Elasticity tensor in the current configuration  objective rates 
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08.10. Elasticity tensor in the current configuration  objective rates 
Krishna Garikipati


08.11. Objectivity of constitutive relations for viscous fluids 
Krishna Garikipati


08.12. Models of viscous fluids 
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08.12. Response to a question 
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08.13. Summary of initial and boundary value problems of continuum mechanics 
Krishna Garikipati


08.14. An initial and boundary value problem of fluid mechanics  the Navier Stokes equations  I 
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08.15. An initial and boundary value problem of fluid mechanics  the Navier Stokes equation  I 
Krishna Garikipati


08.16. An initial and boundary value problem of fluid mechanics  II 
Krishna Garikipati


08.17. Material symmetry 1Isotropy 
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08.17. Response to a question 
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08.18. Material symmetry 2Isotropy 
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08.19. Material symmetry 2Isotropy 
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08.20. Material symmetry 3Isotropy 
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09.01. A boundary value problem in nonlinear elasticity I 
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09.02. A boundary value problem in nonlinear elasticityI 
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09.02. Response to a question 
Krishna Garikipati


09.03. A boundary value problem in nonlinear elasticityII. The inverse method 
Krishna Garikipati


09.03. Response to another question 
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10.01. Linearized elasticityI 
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10.02. Linearized elasticityI 
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10.03. Linearized elasticityII 
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10.04. Linearized elasticityII 
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10.04. Response to a question 
Krishna Garikipati


10.05. Classical continuum mechanics: Books, and the road ahead 
Krishna Garikipati


11.01. The first law of thermodynamicsthe balance of energy 
Krishna Garikipati


11.02. The first law of thermodynamicsthe balance of energy 
Krishna Garikipati


11.03. The first law of thermodynamicsthe balance of energy 
Krishna Garikipati


11.04. The second law of thermodynamicsthe entropy inequality 
Krishna Garikipati


11.05. Legendre transformsthe Helmholtz potential 
Krishna Garikipati


11.06. The ClausiusPlanck inequality 
Krishna Garikipati


11.07. Response to a question 
Krishna Garikipati


11.07. The ClausiusDuhem inequality 
Krishna Garikipati


11.08. The heat transport equation 
Krishna Garikipati


11.09. Thermoelasticity 
Krishna Garikipati


11.10. The heat flux vector in the reference configuration 
Krishna Garikipati


12.01. The free energy functional 
Krishna Garikipati


12.02. The free energy functional 
Krishna Garikipati


12.03. Extremization of the free energy functionalvariational derivatives 
Krishna Garikipati


12.04. EulerLagrange equations corresponding to the free energy functional 
Krishna Garikipati


12.05. The weak form and strong form of nonlinear elasticity 
Krishna Garikipati


12.06. The weak form and strong form of nonlinear elasticity 
Krishna Garikipati


13.01. The setting for mass transport 
Krishna Garikipati


13.02. The setting for mass transport 
Krishna Garikipati


13.03. AsideA unified treatment of boundary conditions 
Krishna Garikipati


13.04. The chemical potential 
Krishna Garikipati


13.05. The chemical potential 
Krishna Garikipati


13.06. Phase separationnon convex free energy 
Krishna Garikipati


13.07. Phase separationnon convex free energy 
Krishna Garikipati


13.08. The role of interfacial free energy 
Krishna Garikipati


13.09. The CahnHilliard formulation 
Krishna Garikipati


13.10. The CahnHilliard formulation 
Krishna Garikipati

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01.01. Introduction 
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01.01. Response to a question 
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01.02. Response to a question 
Krishna Garikipati


01.02. Vectors I 
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01.03. Vectors II 
Krishna Garikipati


01.04. Vectors III 
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02.01. Tensors I 
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02.02. Response to a question 
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02.02. Tensors II 
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02.03. Tensors III 
Krishna Garikipati


02.04. Tensor properties I 
Krishna Garikipati


02.05. Tensor properties I 
Krishna Garikipati


02.06. Tensor properties II 
Krishna Garikipati


02.07. Tensor properties II 
Krishna Garikipati


02.08. Tensor properties III 
Krishna Garikipati


02.09. Vector and tensor fields 
Krishna Garikipati


02.10. Vector and tensor fields 
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03.01. Configurations 
Krishna Garikipati


03.02. Configurations 
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03.03. Motion 
Krishna Garikipati


03.03. Response to a followup question 
Krishna Garikipati


03.03. Response to a question 
Krishna Garikipati


03.04. The Lagrangian description of motion 
Krishna Garikipati


03.05. The Lagrangian description of motion 
Krishna Garikipati


03.06. The Eulerian description of motion 
Krishna Garikipati


03.07. The Eulerian description of motion 
Krishna Garikipati


03.08. The material time derivative 
Krishna Garikipati


03.09. Response to a question 
Krishna Garikipati


03.09. The material time derivative 
Krishna Garikipati

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04.01. The deformation gradient: mapping of curves 
Krishna Garikipati


04.02. The deformation gradient: mapping of surfaces and volumes 
Krishna Garikipati


04.03. The deformation gradient: mapping of surfaces and volumes 
Krishna Garikipati


04.04. The deformation gradient: a firstorder approximation of the deformation 
Krishna Garikipati


04.05. Stretch and strain tensors 
Krishna Garikipati


04.06. Response to a question 
Krishna Garikipati


04.06. Stretch and strain tensors 
Krishna Garikipati


04.07. The polar decomposition I 
Krishna Garikipati


04.08. The polar decomposition I 
Krishna Garikipati


04.09. The polar decomposition II 
Krishna Garikipati


04.10. Response to a question 
Krishna Garikipati


04.10. The polar decomposition II 
Krishna Garikipati


04.11. Velocity gradients, and rates of deformation 
Krishna Garikipati


04.12. Velocity gradients, and rates of deformation 
Krishna Garikipati

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05.01. Balance of mass I 
Krishna Garikipati


05.02. Balance of mass I 
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05.03. Balance of mass II 
Krishna Garikipati


05.04. Balance of mass II 
Krishna Garikipati


05.05. Reynolds' transport theorem I 
Krishna Garikipati


05.06. Reynolds' transport theorem I 
Krishna Garikipati


05.07. Reynolds' transport theorem II 
Krishna Garikipati


05.08. Response to a question 
Krishna Garikipati


05.08. Reynolds' transport theorem III 
Krishna Garikipati


05.09. Linear and angular momentum I 
Krishna Garikipati


05.09. Response to a question 
Krishna Garikipati


05.10. Linear and angular momentum II 
Krishna Garikipati


05.11. The moment of inertia tensor 
Krishna Garikipati


05.12. The moment of inertia tensor 
Krishna Garikipati


05.13. The rate of change of angular momentum 
Krishna Garikipati


05.14. The balance of linear and angular momentum for deformable, continuum bodies 
Krishna Garikipati


05.15. The balance of linear and angular momentum for deformable, continuum bodies 
Krishna Garikipati


05.16. The Cauchy stress tensor 
Krishna Garikipati


05.17. Stress  An Introduction 
Krishna Garikipati

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06.01. Balance of energy 
Krishna Garikipati


06.01. Response to a followup question 
Krishna Garikipati


06.01. Response to a question 
Krishna Garikipati


06.02. Additional measures of stress 
Krishna Garikipati


06.03. Additional measures of stress 
Krishna Garikipati


06.03. Response to a followup question 
Krishna Garikipati


06.03. Response to a question 
Krishna Garikipati


06.04. Work conjugate forms 
Krishna Garikipati


06.05. Balance of linear momentum in the reference configuration 
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07.01. Equations and unknowns  constitutive relations 
Krishna Garikipati


07.01. Response to a question 
Krishna Garikipati


07.02. Constitutitve equations 
Krishna Garikipati


07.03. Elastic solids and fluids  hyperelastic solids 
Krishna Garikipati


07.03. Response to a question 
Krishna Garikipati

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08.01. Objectivity  change of observer 
Krishna Garikipati


08.02. Objectivity  change of observer 
Krishna Garikipati


08.03. Objective tensors, and objective constitutive relations 
Krishna Garikipati


08.04. Objective tensors, and objective constitutive relations 
Krishna Garikipati


08.05. Objectivity of hyperelastic strain energy density functions 
Krishna Garikipati


08.06. Examples of hyperelastic strain energy density functions 
Krishna Garikipati


08.07. Examples of hyperelastic strain energy density functions 
Krishna Garikipati


08.07. Response to a question 
Krishna Garikipati


08.08. The elasticity tensor in the reference configuration 
Krishna Garikipati


08.09. Elasticity tensor in the current configuration  objective rates 
Krishna Garikipati


08.10. Elasticity tensor in the current configuration  objective rates 
Krishna Garikipati


08.11. Objectivity of constitutive relations for viscous fluids 
Krishna Garikipati


08.12. Models of viscous fluids 
Krishna Garikipati


08.12. Response to a question 
Krishna Garikipati


08.13. Summary of initial and boundary value problems of continuum mechanics 
Krishna Garikipati


08.14. An initial and boundary value problem of fluid mechanics  the Navier Stokes equations  I 
Krishna Garikipati


08.15. An initial and boundary value problem of fluid mechanics  the Navier Stokes equation  I 
Krishna Garikipati


08.16. An initial and boundary value problem of fluid mechanics  II 
Krishna Garikipati


08.17. Material symmetry 1Isotropy 
Krishna Garikipati


08.17. Response to a question 
Krishna Garikipati


08.18. Material symmetry 2Isotropy 
Krishna Garikipati


08.19. Material symmetry 2Isotropy 
Krishna Garikipati


08.20. Material symmetry 3Isotropy 
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09.01. A boundary value problem in nonlinear elasticity I 
Krishna Garikipati


09.02. A boundary value problem in nonlinear elasticityI 
Krishna Garikipati


09.02. Response to a question 
Krishna Garikipati


09.03. A boundary value problem in nonlinear elasticityII. The inverse method 
Krishna Garikipati


09.03. Response to another question 
Krishna Garikipati

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10.01. Linearized elasticityI 
Krishna Garikipati


10.02. Linearized elasticityI 
Krishna Garikipati


10.03. Linearized elasticityII 
Krishna Garikipati


10.04. Linearized elasticityII 
Krishna Garikipati


10.04. Response to a question 
Krishna Garikipati


10.05. Classical continuum mechanics: Books, and the road ahead 
Krishna Garikipati

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11.01. The first law of thermodynamicsthe balance of energy 
Krishna Garikipati


11.02. The first law of thermodynamicsthe balance of energy 
Krishna Garikipati


11.03. The first law of thermodynamicsthe balance of energy 
Krishna Garikipati


11.04. The second law of thermodynamicsthe entropy inequality 
Krishna Garikipati


11.05. Legendre transformsthe Helmholtz potential 
Krishna Garikipati


11.06. The ClausiusPlanck inequality 
Krishna Garikipati


11.07. Response to a question 
Krishna Garikipati


11.07. The ClausiusDuhem inequality 
Krishna Garikipati


11.08. The heat transport equation 
Krishna Garikipati


11.09. Thermoelasticity 
Krishna Garikipati


11.10. The heat flux vector in the reference configuration 
Krishna Garikipati

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12.01. The free energy functional 
Krishna Garikipati


12.02. The free energy functional 
Krishna Garikipati


12.03. Extremization of the free energy functionalvariational derivatives 
Krishna Garikipati


12.04. EulerLagrange equations corresponding to the free energy functional 
Krishna Garikipati


12.05. The weak form and strong form of nonlinear elasticity 
Krishna Garikipati


12.06. The weak form and strong form of nonlinear elasticity 
Krishna Garikipati

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13.01. The setting for mass transport 
Krishna Garikipati


13.02. The setting for mass transport 
Krishna Garikipati


13.03. AsideA unified treatment of boundary conditions 
Krishna Garikipati


13.04. The chemical potential 
Krishna Garikipati


13.05. The chemical potential 
Krishna Garikipati


13.06. Phase separationnon convex free energy 
Krishna Garikipati


13.07. Phase separationnon convex free energy 
Krishna Garikipati


13.08. The role of interfacial free energy 
Krishna Garikipati


13.09. The CahnHilliard formulation 
Krishna Garikipati


13.10. The CahnHilliard formulation 
Krishna Garikipati
