Lecture Description
- The CosmoLearning Team
Course Index
- Introduction
- Response to a question
- Vectors I
- Response to a question
- Vectors II
- Vectors III
- Tensors I
- Tensors II
- Response to a question
- Tensors III
- Tensor properties I
- Tensor properties I
- Tensor properties II
- Tensor properties II
- Tensor properties III
- Vector and tensor fields
- Vector and tensor fields
- Configurations
- Configurations
- Motion
- Response to a question
- Response to a follow up question
- The Lagrangian description of motion
- The Lagrangian description of motion
- The Eulerian description of motion
- The Eulerian description of motion
- The material time derivative
- The material time derivative
- Response to a question
- The deformation gradient: mapping of curves
- The deformation gradient: mapping of surfaces and volumes
- The deformation gradient: mapping of surfaces and volumes
- The deformation gradient: a first order approximation of the deformation
- Stretch and strain tensors
- Stretch and strain tensors
- The polar decomposition I
- Response to a question
- The polar decomposition I
- The polar decomposition II
- The polar decomposition II
- Velocity gradients, and rates of deformation
- Response to a question
- Velocity gradients, and rates of deformation
- Balance of mass I
- Balance of mass I
- Balance of mass II
- Balance of mass II
- Reynolds' transport theorem I
- Reynolds' transport theorem I
- Reynolds' transport theorem II
- Reynolds' transport theorem III
- Response to a question
- Linear and angular momentum I
- Correction to boardwork
- Response to a question
- Linear and angular momentum II
- The moment of inertia tensor
- The moment of inertia tensor
- The rate of change of angular momentum
- The balance of linear and angular momentum for deformable, continuum bodies
- The balance of linear and angular momentum for deformable, continuum bodies
- The Cauchy stress tensor
- Stress-- An Introduction
- Balance of energy
- Response to a question
- Response to a follow up question
- Additional measures of stress
- Additional measures of stress
- Response to a question
- Response to a follow up question
- Work conjugate forms
- Balance of linear momentum in the reference configuration
- Equations and unknowns--constitutive relations
- Response to a question
- Constitutitve equations
- Elastic solids and fluids--hyperelastic solids
- Response to a question
- Objectivity--change of observer
- Objectivity--change of observer
- Objective tensors, and objective constitutive relations
- Objective tensors, and objective constitutive relations
- Objectivity of hyperelastic strain energy density functions
- Examples of hyperelastic strain energy density functions
- Examples of hyperelastic strain energy density functions
- Response to a question
- The elasticity tensor in the reference configuration
- Elasticity tensor in the current configuration--objective rates
- Elasticity tensor in the current configuration--objective rates
- Objectivity of constitutive relations for viscous fluids
- Models of viscous fluids
- Response to a question
- Summary of initial and boundary value problems of continuum mechanics
- An initial and boundary value problem of fluid mechanics--the Navier Stokes equations
- An initial and boundary value problem of fluid mechanics--the Navier Stokes equation
- An initial and boundary value problem of fluid mechanics II
- Material symmetry 1--Isotropy
- Response to a question
- Material symmetry 2--Isotropy
- Material symmetry 2--Isotropy
- Material symmetry 3--Isotropy
- A boundary value problem in nonlinear elasticity I
- A boundary value problem in nonlinear elasticity I
- Response to a question
- A boundary value problem in nonlinear elasticity II--The inverse method
- Response to another question
- Linearized elasticity I
- Linearized elasticity I
- Linearized elasticity II
- Linearized elasticity II
- Response to a question
- Classical continuum mechanics: Books, and the road ahead
- The first law of thermodynamics the balance of energy
- The first law of thermodynamics the balance of energy
- The first law of thermodynamics the balance of energy
- The second law of thermodynamics the entropy inequality
- Legendre transforms the Helmholtz potential
- The Clausius Planck inequality
- The Clausius Duhem inequality
- Response to a question
- The heat transport equation
- Thermoelasticity
- The heat flux vector in the reference configuration
- The free energy functional
- The free energy functional
- Extremization of the free energy functional variational derivatives
- Euler Lagrange equations corresponding to the free energy functional
- The weak form and strong form of nonlinear elasticity
- The weak form and strong form of nonlinear elasticity
- The setting for mass transport
- The setting for mass transport
- Aside A unified treatment of boundary conditions
- The chemical potential
- The chemical potential
- Phase separation non convex free energy
- Phase separation non convex free energy
- The role of interfacial free energy
- The Cahn Hilliard formulation
- The Cahn Hilliard formulation
Course Description
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