Overview. This post is a description of the project Sunny Wicks and I will be pursuing for this class, to satisfy the requirements for Problem 29.
ES 240 Problem Set #8, Problem #20 - Green's function of biharmonic operator is not positive definiteSubmitted by John Peter Whitney on Thu, 2007-12-06 23:44.
Professor Vlassak mentioned that last year every single person did a finite element project. He said he wanted to see more theory projects, so I decided to take him up on that.
I was browsing around one day and happened upon an article that explained that while the Green's function of the laplacian was positive definite, the biharmonic operator's Green's function is not. Physically, this has significance.
In a previous post, Learning to be a PhD advisor, I wrote about learning to do my job from students. Over the years, I have also learned from writings of other scientists on doing research, its dynamics: competition, despair, and exhilaration... Here is a small sample that occurs to me this morning.
See attachment for ES 240 lecture notes on viscoelasticity.
This problem set is due Friday, Dec.7, 2007.
ES 240 notes for Bending of plates is attached.
This problem set is due Monday, Nov.26, 2007.
This computer assignment is due Friday, Nov. 16, 2007
ES 240 notes for Principle of virtual work and FEM. Please see attached.
I recommend this classic book to persons, who want to learn something more about elasticity that cannot be found in traditional books. It's a pretty valuable and inspiring book in elasticity today, although it was written by Love more than 100 years ago. It included many impressive topics such as equilibirum of anistropic elastic solid bodies, the equilibrium of a elastic sphere, plates and shells. As far as I know, this book is frequently quoted in recent artilces. However, it will be very tough to read this book, even though you have some basic knowledge about elasticity. I have only scanned some chapters. Everyone should have a try.
This book begins by describing real life examples of mechanical states of different materials. The book next discusses stress. This discussion includes force, mohr circles, tensor components of stress, and stress fields. Next strain is discussed. This ranges from measuring deformation to tensor components of infinite and finite strain. The book concludes by outlining different material behaviors. These include Hookean behavior and Newtonian behavior. This last section also discusses energy consumed in deformation.
This book presents material in the same sequence as it is discussed in class, but with more attention to details. This helps to fill in the gaps for things that students might miss during the lectures.
Title: Theory of Plates and Shells
Author: Stephen P. Tomoshenko and S. Woinowsky-Krieger
Chapter 1: Bending of long rectangular plates to a cylindrical surface .
Chapter 2: Pure bending of plates.
Chapter 3: Symmetrical bending of circular plates
Chapter 4: Small deflections of laterally loaded plates
Chapter 5: Simply supported rectangular plates
Chapter 6: Rectangular plates with various edge conditions
Chapter 7: Continuous rectangular plates
Chapter 8: Plates on elastic foundation
Chapter 9: Plates of various shapes
Chapter 10: Special and approximate methods in theory of plates
This problem set is due on Nov.2, 2007.
Theory of Elasticity by Landau and Lifshitz.
1 fundamental equations
2 the equilibrium of rods and plates
3 elastic waves
5 thermal conduction and viscosity in solids
6 mechanics of liquid crystals
If I were to recommend one textbook that will help students in this course it would obviously be the "Theory of Elasticity" by Timoshenko and Goodier. But you could have found that out by simply looking at the course syllabus, so I will also recommend the following books that are helpful in other areas of the course: "Mathematical Phyiscs" by Kusse and Westwig, "Mechanics of Materials" by Beer and Johnson, and "Advanced Engineering Mathematics" by Greenberg.
There are a few helpful reviews of this book on amazon. Here is the link to that page:
The hyperlink for reviews on this book on amazon.com is the following:
Mechanics of Materials was used as the textbook in my undergraduate solid mechanics course. It is an introductory book which gives a great overview of the basic concepts needed for solid mechanics . The material is presented in a way that makes it easy to understand with many practical examples. I learn material best when I am shown how theories are applied and this book does that very well. It also dives into detail of some practical applications of fundamental solid mechanics. The book explores axial loading, torsion, pure bending, analysis of beams, shearing stresses in beams, transformations, principal stresses, deflection of beams, columns, and energy methonds.
This book may not seem like it would help in this course very much, but I used it for the problem set that we had on the compliance and stiffness matrices. The portions of the text that pertain to this course are Chapters 4 and 5. Chapter 4 is about tensors. This chapter also includes a review of suffix notation with dummy indicies. Chapter 5 is about stress, strain and elasticity. I used this chapter for better understanding of the tensor notation and to see how to work with the stiffness and compliance matrices. The rest of the text is about crystal structures, defects in crystals, and transformations of crystals. But Chapters 4 and 5 have helped me.
Deformable Bodies and Their Material Behavior by HW Haslach and RW Armstrong is a great reference book for solid mechanics. This text discusses a wide variety of materials, the relationships between applied stresses, displacements and material properties, the mathematical approximations to predict mechanical behaviors, and the practical uses for the theory. The text helps to understand how the theory can be applied to practical problems. The text has many worked examples to common problems.
I recommend the book
“The Linearized Theory of Elasticity” by William S.
Here is a review of it from Amazon:
1 Review of Mechanics of Materials
Outline of Content:
I find the book , An Introduction to the mechanics of solids , is very helpful to me.
This book offers detailed discussion on modeling, placing emphasis on where the equations come from and why some variable should be zero or can be ignored. Thus I can learn not only the derivation but also the mechanical insight.
Amazon review link:
Each year, several new students begin their studies of Solid Mechanics for PhD degrees at Harvard School of Engineering and Applied Sciences. The students come from all over the world. We have no constraint on where they come from.
Faculty members in Solid Mechanics. The School of Engineering and Applied Sciences is not divided into departments, but faculty do assemble into programs. Faculty members directly responsible for the program of Solid Mechanics are