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User:Jacob J. Walker/iMet Cohort 11 – Statement of Purpose

Personal Information

 * Name: Jacob Walker
 * Email: [mailto:JacobWalker@grant.k12.ca.us JacobWalker@grant.k12.ca.us]
 * Current Employment: Grant Adult Education (Grant Joint Union High School District)

What are your life goals for the near future?
I currently am a computer instructor with Grant Adult Education, and plan to stay an instructor here for about 3 to 5 more years. I am simultaneously starting to work on the Effective Education Project (www.effectiveeducation.org), which has the ambitious (but doable) goal of creating an improved educational system from the ground up. This system will first be implemented in nations where people are at a level close to survival, and will assist people in improving their lives and the lives of those around them. This may become a full time career or it may stay as an equal project to my employment.

After finishing my Masters, I will probably work towards teaching in a program like iMet, or the BVE program, or some other progam where I'm helping to teach teachers. Beyond this, I have considered possibly joining the Peace Corp and/or possibly even applying to the Sloan School of Management at MIT to enter their Ph.D. program in Organizational Behavior.

How do you think the iMet Program might help you attain these goals?
Dr. John Cowan talked about how everyone has something to gain from the iMet program. We all have things to learn. I agree with this statement. I have struggled internally with justifying to myself about spending the money to get “a piece of paper”. But I believe the iMet will give me more than just a piece of paper. I am interested in working with a group of people who are dedicated to education, and who are at a higher level, working to achieve more. I truly believe the best ideas come from collaboration, compared to what we can generate on our own, no matter how intelligent or how much experience we have, we can always miss something. Jack Welch, arguably one of the most important CEOs of the 20th century, would purposely seek to talk with people from all walks of life and many perspectives, even those not directly involved with his business.

While I suspect that I already know much of the theory that will be taught in the classes, the iMet program will help fill in some gaps, reinforce some ideas, and assist me to dig deeper into certain ideas and see how they can really be implemented to improve education. Further, being part of a cohort of intelligent, passionate, and technologically-interested people will be excellent in helping to develop some of these ideas. With collaboration we can discover what ideas make sense, which ones don’t, and how to improve upon all of them.

The iMet Masters will of course also help me improve my current pay scale and give me more respect in the eyes of my peers and people I will be working with for the Effective Education Project. It also will allow me to teach at the college or university level in the future. But if these were the only reasons that I was applying, I would not apply. It is because I believe that the iMet process has something to give to me, and I have something to give to it, that I wish to join.

What role do you believe technology should play in K-12 or other education/training settings?
To understand what role technology can and should play in education, it is first important to understand technology. Any piece of technology is both a tool and a system. We usually think of computers as synonymous with technology, but in fact technology truly includes all human designed physical systems, including the buses that transport kids to school. And even the term "Information Technology" should be understood to include books, and the production of books, which is one of the older forms of technology used in education. While it is important to recognize the breadth of technology, I will limit my discussion here to that of computer based information systems.

Computers have the potential to truly benefit education. But during this discussion about how they can be of benefit it is important to attempt to be aware of the unintended consequences that any tool (technology) may have, and to use this understanding while designing an educational system.

One of the first unintended consequences that need to be considered with any technology can be summed by quoting Abraham Maslow: "If the only tool you have is a hammer, you tend to see every problem as a nail." Often school administrators and technology zealots see computers as a cure all to all education problems. They start to see every problem as something a computer can solve. While a computer is a tool that can probably assist in the solution to most problems, it is not the only tool that should be considered.

Another unintended consequence comes from Marshall McLuhan's observation that any extension of man (technology) amputates other extensions. . By this it is important to understand that if students only use the Internet or computers to try and understand the world, they may not use their natural senses as often to simply walk into a forest and observe the animals and plants. Since there is only a limited amount of time that an education system has to expose any learner in the system to different things, careful thought needs to be made in the engineering of the educational system to maximize all aspects of what a learner truly gains.

Cost to Benefit must also always be considered with the use of technology. As an example, I generally don't believe that SmartBoards are worth the extra expense to the educational gains that may be made by using them. Other technological solutions may be able to accomplish the same goals, nearly as effectively for less cost.

Yet, even given these and similar issues, computer technology has a lot that it can do for education. Combined with the Internet, computers can store and transmit information to a higher degree than ever possible before. This gives students the ability to gain information from many other locations than ever before. Computers also allow students to interact with information in different ways than previously. Utilized in a Constructivist fashion, technology like spreadsheets allow students to be able to use math as a manipulative, and if the educational system is engineered properly, this technology can also help students learn how math can be utilized. An example of this is being developed in the Public Pupil Pizza Project.

Every day in my classes, I use technology to help teach technology. I use LCD projectors so my class can see and follow along with the steps I am doing. I use the printer to make handouts that they can use later to review the lesson. I teach them how to use Windows, Word, Excel, Access and PowerPoint. But with each one of these, I am moving towards a curriculum of applied academics. For example, I teach my students how to use Word and Publisher to make resumes, resume "business cards", cover letters, portfolios, and other job seeking tools. I show how Excel can be used in real life situations, and am working towards fully developing it with the Public Pupil Pizza Project. I teach my students how to do Business Intelligence with Access where they learn how to retrieve relevant information from a database and then how to analyze and use that information to improve an organization. These are all tangible things that technology has had a role in both assisting in education and being the content of the education.

What role do you think technology should play in creating a democratic and pluralistic world?
First, it is important to think about what it takes to create a democratic and pluralistic world. America, for all of its faults, is greatly ahead of much of the world in having an open society, democracy, wealth, and acceptance. As citizens of planet earth, if we truly believe that all humans were created equal, as our Declaration of Independence suggests, then we logically should focus on our human brothers and sisters who are in the most need. These are NOT the people in the United States. Even the poorest among us, who live on the streets, can make more money in one hour begging, than most of the world can in one day of hard work.

Also, before answering this question we must realize that technology is always a "double edged sword". It can be used for "good" or for "bad". As educators, I believe it is our ethical duty to attempt to make sure that what we teach will benefit the world. So while we are teaching about technology we also must be teaching about developing a system of ethics, and challenge learners to think about the consequences technology may have.

With that being said, computer technology has been shown to be an equalizing force in the world. Wikipedia is available to anyone who can access the Internet; it also gives the entire world an open marketplace of ideas where generally the best and most accurate ideas win, and a fuller picture is developed on any subject than within a traditional encyclopedia. India has been able to improve the welfare of many of its highly educated citizens by allowing them to get high tech jobs that were once only available in America or other "developed" nations. And thus has also forced America to compete in the global marketplace of ideas, and not be able to sit on our laurels. Thomas Friedman discusses much of this in his excellent book "The World is Flat"

With the Effective Education Project, I am working with Muhsin Jallal and the Northern Volunteers Group in Ghana to build a computer school there that they can earn money by doing transcription. We are working to develop a system where organizations in the U.S. and other developed nations who produce audio, such as college lectures, church services, talk radio, etc, can have this audio transcribed to text. You can find out more information about this in my article about Economic Leap Frogging through Trading Information via the Internet

How do you feel about joining a cohort program that requires that you make a commitment to teamwork and community?
I think the cohort has the potential to be the most positive aspect about the iMet program for me. Many of the ideas that I have started to develop, will not be able to grow by just me alone. It will take collaboration with people who are passionate about similar goals, and who are capable of helping to make these goals a reality. Being part of a tight cohort will give me the opportunity to express my ideas to people who are potentially within that subset. I hope that I will find others who can see the importance of the ideas I started, and further to be able to argue the advantages and disadvantages of different ideas, and be able to test the ideas through action research. I also hope that the ideas that I'm working will gain "co-ownership" from others within the group, so they are not "my" ideas, but end up being "our" ideas. The only way a cohort might be a negative experience for me, is I could not find people of similar enough viewpoint who would be interested in working with me on the projects that I'm passionate about.

What do you think might be the advantages to learning in an online environment?
For myself, in the iMet program, being in an online environment can allow for more flexibility. Asynchronous communication allows me to have flexibility with my time, and even synchronous communication will allow me to have flexibility with my location.

An online environment has other potential advantages for learners, which is the fact that they don't need to be geographically close to a teacher. This will be important to me as I work more with the Effective Education Project, because it will probably generally have a better benefit to cost ratio to do online training with students in less developed nations than with attempting to fly out to each country.

What do you think might be the challenges?
There are several challenges of online education. One comes from the medium of being "online" and the other comes with the benefit of flexibility.

As I had discussed earlier about Marshall McLuhan's perspective that one "extension of man" can amputate another, having only online education amputates many of the benefits that in person communication and schooling has. This is why I'm very glad that the iMet program still has the 25% component of being in person, and especially happy that it has the initial retreat. To me that seems to be a far more well designed educational system than what many universities are doing with online only education.

The inherent advantage of flexibility that is gained through using online education, requires self discipline and dedication to not have it become a problem. Procrastination is a common issue among people, and one that I've had problems with most of my life. Although, if I am passionate about a task, and take the time to plan the group of tasks ahead of time, and truly dedicate myself to them, I have shown that I can make great accomplishments. So I feel I am up to the challenge.

The issue of working with online education with people in other countries often has the additional challenge of making sure someone understands you, especially when English is not their native language. It also is more difficult to determine how they are progressing in tangible things. As the Effective Education Project gets developed more, these issues and others will need to find methods of being resolved.

Is there anything else you would like to add?
I think that I have written enough so far. Any of the topics I have thus far written can be expanded upon, and I am sure I will continue to learn more about each of them. I wouldn't be entering the iMet if I didn't feel I could or would learn more.


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User talk:Vainentree

Toodles.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: User talk:Vainentree


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User:Raju~enwikiversity

I completed and got my Engineering degree in Electrical and Electronics; and currently working as a Web Developer in a software company for last 2 years.

I love Blogging and websites. See my website at Raju Sykam's Home page and my blog on Electrical n Electronics and Computers Raju Sykam's blog

title: User:Raju~enwikiversity


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User:WODUP

See http://en.wikipedia.org/wiki/User:WODUP.

title: User:WODUP


text: Template:Mechanical engineering/box-header


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Portal:Mechanical engineering/Intro

Mechanical engineering is a very broad field that involves the application of physical principles for analysis, design, manufacturing, and maintenance of mechanical systems. It is made up of a number of subdisciplines concerned with the mechanics, kinematics (movement), and energy of physical objects. Practitioners of mechanical engineering, known as mechanical engineers, use principles such as heat, force, and the conservation of mass and energy in contributing to the design of vehicles and aircraft, heating & cooling systems, buildings and bridges, industrial equipment and machinery, and much more.

title: Portal:Mechanical engineering/Intro


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Portal:Mechanical engineering/Subdivision

Like divisions, subdivisions of Mechanical Engineering are pages in the (Topic namespace) and their names start with the "Topic:" prefix. Some subdivisions of Mechanical Engineering such as Applied Mechanics can be shared by multiple departments both in Engineering and by multiple schools. Schools that use (link to) the same department should cooperate to develop the department. If you want to add another subdivision to Mechanical Engineering, you can use the Template:Subdivision boilerplate template to start a subdivision.lists are given in the topic. So you can learn more over from the bottom to top
 * Applied Mechanics
 * Design and Manufacturing
 * Dynamics and Control
 * Thermal Sciences
 * Interdisciplinary Studies


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Portal:Mechanical engineering/Subjects

You can add links to pages on various subject areas here. However, such pages should ultimately be moved to the appropriate subdivision of mechanical engineering. To add additional subject pages to this division, edit the Mechanical engineering template. Other subdivisions should have their associated templates. You can also use a template to start off a new page. For example, if you want to use a subdivision boilerplate then type into the edit box and press save.

title: Portal:Mechanical engineering/Subjects


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Portal:Mechanical engineering/Featured learning resource

 * Introduction to elasticity
 * Continuum mechanics
 * Introduction to finite elements
 * Nonlinear finite elements
 * Abaqus help desk


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Portal:Mechanical engineering/Division news

 * December 09, 2010 Member Kalumajhi joined the division!
 * December 17, 2007 Member phoenixnem joined the division!
 * October 11, 2007 Member gustable joined the division!
 * August 4, 2007 Division page given a new look.
 * October 11, 2006 Member Wahib Tannoury joined the division!
 * October 05, 2006 Member Hawwari joined the division!
 * October 03, 2006 Member navarai joined the Division!
 * September 25, 2006 Member Moo7a joined the Division!
 * August 22, 2006 - Division founded!


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Portal:Mechanical engineering/Participants

 * Moo7a 01:21, 25 September 2006 (UTC)
 * cvasques 17:34, 9 February 2008 (UTC)
 * Jamesontai 16:57, 1 October 2008 (UTC)


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Portal:Mechanical engineering/Mechanical engineering External links

 * American Society Of Mechanical Engineers
 * Mechanical Engineering Portal
 * Free Online Course Materials from Massachusetts Institute of Technology
 * iMechanica - the web of mechanicians


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Template:Mechanical engineering/Research projects

This feature of Wikiversity will be implemented later pending further discussion.


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Portal:Mechanical engineering/Quotes

"In the workplace, employees are not in competition with each other but have to work together to get the product out the door. Those mechanical students who understand this from the get-go will go a lot further." -Charlotte Thomas (from "From Mechatronics to Medicine, Mechanical Engineers Push Employers' Hot Buttons", Graduating Engineer, 2007)

"An expert is a man who has made all the mistakes, which can be made, in a very narrow field." —Niels Bohr

title: Portal:Mechanical engineering/Quotes


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Portal:Mechanical engineering/Learning projects

 * Introduction to Elasticity
 * Continuum mechanics
 * Introduction to finite elements
 * Nonlinear finite elements
 * Abaqus help desk


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Portal:Mechanical engineering/WikiProjects

Please list your content development projects related to Mechanical engineering here.

Lunar Boom Town provides opportunities for engineers to help analyze technologies and venture approachs in conjunction with entrepreneurs, customers, and other interested parties.


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Template:Mechanical engineering/Things you can do

 * Add your class notes to the relevant subject page.

 * Do an applicable research paper and publish it here after it is graded.


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File:Ward+video.ogg

Summary
This OGG video file includes an audio clip from a WikipediaWeekly interview of Ward Cunningham.

Copyleft photo credits; images from Wikimedia Commons

 * User:ed_g2s  Image:WikiMeet London - Fuzheado.jpg
 * Image:Andrew Lih - Wikimania 2006 - Hacking Day 2 at OLPC.jpg By Andrew Lih
 * Image:WM2007 Opening Jimbo.jpg Author= Rico Shen
 * User:KaurJmeb Image:Wm07wikiball.JPG

title: File:Ward+video.ogg


text: Portal:Applied Mechanics/box-header

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User:JWSchmidt/Blog/4 August 2007

I'm interested in having a media-rich environment for Wikiversity.

Wikimania
During day 2 of Wikimania there was streaming video of the meeting sessions. The WikipediaWeekly team has also been doing podcasts from Taiwan. I was pleased to hear Ward Cunningham interviewed on a podcast and talking about the idea of extending online collaborations beyond the written word into collaborative creation of videos.

Collaborative creation of video
I cut out the section of the WikipediaWeekly interview with Ward Cunningham that has what he said about the idea of a transition from online wiki collaborations with a focus on creating documents like encyclopedia articles towards new possibilities such as collaborative creation of video resources. The audio clip from the interview starts with Ward talking about how wiki technology was in a "sweet spot" when created in 1995, but maybe computer technology is now at the point where wiki-like collaboration can be extended beyond just words and into video. One approach mentioned by Ward is the use of Machinima and using video game-like tools to create video.

Wikiversity: beyond words
One of my personal goals for Wikiversity has been to make it possible for Wikiversity participants to move beyond a conventional wiki with words and a few images towards an environment that supports rich multimedia learning resources. Some relevant pages are:
 * Wikiversity the Movie
 * Wikiversity Reports
 * Rich Learning Resources Platform
 * Animation with DAZ Studio
 * Wiki Campus Radio
 * Web 2.0
 * Sandbox Server

Content Creators
I hope Wikiversity can be a learning center for helping people become full participants in the Web 2.0 revolution and the Free culture movement. Wikiversity should be able to play an important role in helping people become internet content creators rather than just content consumers.

Software development

 * Wikipedia Invites Users to Take Part in Open, Collaborative Video Experiment - press release, January 17, 2008.

title: User:JWSchmidt/Blog/4 August 2007


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Template:Applied Mechanics/Intro

Applied mechanics is a broad topic that is a subset of the activities of mechanical engineers, aeronautical engineers, bioengineers, materials scientists, mining engineers, naval engineers, geophysicts, physicists, mathematicians, and many other types of engineers and scientists.

title: Template:Applied Mechanics/Intro


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Portal:Applied Mechanics/Departments

Since applied mechanics is a truly interdisciplinary subject, this subdivision provides links to several departments and is linked from the pages of several departments.

Aeronautical Engineering. Applied Mathematics. Applied Physics. Biomedical Engineering. Geophysics. Geotechnical Engineering. Materials Science and Engineering. Mechanical Engineering. Metallurgical Engineering. Mining Engineering. Ocean Engineering

title: Portal:Applied Mechanics/Departments


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Portal:Applied Mechanics/Subjects

You can add links to pages on various subject areas here. However, such pages should ultimately be moved to the appropriate subdivision of applied mechanics. To add additional subject pages to this division, edit the Applied Mechanics template. Other subdivisions should have their associated templates. You can also use a template to start off a new page. For example, if you want to use a subdivision boilerplate then type into the edit box and press save.

title: Portal:Applied Mechanics/Subjects


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Portal:Applied Mechanics/Subdivisions

"In the application of the natural sciences, mechanics is complemented by thermodynamics, the study of heat and more generally energy, and electromechanics, the study of electricity and magnetism." -- Nobel Laureate Gilbert N. Lewis

To add additional subject pages to this division, edit the Applied Mechanics template and/or the other appropriate templates within.

title: Portal:Applied Mechanics/Subdivisions


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Portal:Applied Mechanics/Featured learning resource

Introduction to Elasticity


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Portal:Applied Mechanics/Division news

 * Saturday, 4 Aug 2007 - Page redesign.
 * Friday, 20 July 2007 - Subdivision founded!

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Portal:Applied Mechanics/Learning projects

 * Solid Mechanics
 * Elasticity
 * Introduction to Elasticity
 * Fluid Mechanics


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Portal:Applied Mechanics/Participants

The histories of Wikiversity pages indicate who the active participants are. If you are an active participant in this subdivision, you can list your name here (this helps small subdivisions grow and participants communicate better; for large subdivisions a list of active participants is not needed). Shuhail
 * Gustable

 * [[User:Shuhail|Shuhail]

 * Savannah

title: Portal:Applied Mechanics/Participants


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Template:Applied Mechanics/Applied Mechanics External links

 * iMechanica - the web of mechanicians
 * MIT Course Materials On-line


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Portal:Applied Mechanics/Discussion forum

 * Please take a look at the talk page!! Gustable 03:58, 9 November 2007 (UTC)
 * 4 Aug 2007 - Need to update courses on introductory statics and dynamics : User:Banerjee


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Template:Applied Mechanics/Things you can do

Add your course notes to the relevant subject areas.


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Portal:Applied Mechanics/Research projects

Coming soon ...


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Portal:Applied Mechanics/WikiProjects

 * Introduction to Elasticity
 * Advanced elasticity
 * Micromechanics of composites
 * Continuum mechanics
 * Introduction to finite elements
 * Nonlinear finite elements
 * Waves in composites and metamaterials
 * Help:Abaqus

title: Portal:Applied Mechanics/WikiProjects


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Portal:Applied Mechanics/Quotes

"A common mistake that people make when trying to design something completely foolproof is to underestimate the ingenuity of complete fools." - Douglas Adams (from Mostly Harmless)

Clifford Truesdell on a paper on isotropic elasticity: "This paper, whose intent is stated in its title, gives wrong solutions to trivial problems. The basic error, however, is not new."

title: Portal:Applied Mechanics/Quotes


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Template:Solid mechanics/Intro

Solid mechanics deals with the deformation, stresses and failure of materials in the solid state.

title: Template:Solid mechanics/Intro


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Portal:Solid mechanics/Departments

Since solid mechanics is a truly interdisciplinary subject, this subdivision provides links to several departments and is linked from the pages of several departments.

Mechanical Engineering. Aeronautical Engineering. Bioengineering. Materials Science and Engineering. Metallurgical Engineering. Mineral Processing. Mining Engineering. Naval Engineering. Ocean Engineering. Geophysics. Applied Mathematics. Applied Physics

title: Portal:Solid mechanics/Departments


text: Portal:Solid mechanics/Subjects

title: Portal:Solid mechanics/Subjects


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Portal:Solid mechanics/Participants

The histories of Wikiversity pages indicate who the active participants are. If you are an active participant in this subdivision, you can list your name here (this can help small subdivisions grow and the participants communicate better; for large subdivisions a list of active participants is not needed).

title: Portal:Solid mechanics/Participants


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User:Open Research/tools

For linking Wikis to Wikis:

 * Interwiki link templates at Wikimedia
 * Interwiki linking at Wikipedia
 * Interwiki link templates at Wikibooks
 * Interwiki link templates at wikimedia
 * Wikimedia help with interwiki linking

title: User:Open Research/tools


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Template:Solid mechanics/Featured learning resource

Please add more as you find or create them.
 * Introduction to Elasticity
 * Introduction to Biomechanics


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Template:Solid mechanics/Learning projects

Introduction to Elasticity


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Portal:Solid mechanics/Division news

 * August 4,2007 - Redesign.
 * July 21, 2007 - Subdivision founded!

title: Portal:Solid mechanics/Division news


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Portal:Solid mechanics/Discussion forum

 * Please add courses on Introductory mechanics of materials (strength of materials).
 * Please add courses on Introductory mechanics of materials (strength of materials).

title: Portal:Solid mechanics/Discussion forum


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Portal:Solid mechanics/Solid mechanics External links

 * iMechanica:The Web of Mechanics and Mechanicians


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Template:Solid mechanics/Things you can do

 * Use your class notes to create learning materials in the appropriate subject areas.


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Portal:Solid mechanics/Research projects

Coming soon ...


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Portal:Solid mechanics/WikiProjects

 * Mechanics
 * Introduction to Elasticity
 * Continuum mechanics
 * Micromechanics of composites
 * Topic: Mechanical vibrations
 * Waves in composites and metamaterials
 * Finite elements
 * Introduction to finite elements
 * Nonlinear finite elements
 * Help:Abaqus

title: Portal:Solid mechanics/WikiProjects


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Portal:Solid mechanics/Quotes

"Opportunity is missed by most people because it is dressed in overalls and looks like work." - Thomas Edison

"Strive for design simplicity, you never have to fix anything you leave out." - Bill Lear

"Always do the things you fear the most; courage is an acquired taste..." - Erica Jong

title: Portal:Solid mechanics/Quotes


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File:3D Storyboard shot 9 needs correction.jpg

Used with the WikiU Film School Lesson on how to create 3D storyboards with FrameForge 3D Studio Demo Version.


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File:3D Storyboard shot 12 needs correction.jpg

Used with the WikiU Film School Lesson on how to create 3D storyboards with FrameForge 3D Studio Demo Version.


text: Template:Elasticity/box-header

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title: Template:Elasticity/box-footer


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Template:Elasticity/Topic news

 * August 4, 2007 - Learning resource Introduction to Elasticity added.
 * July 21, 2007 - Subdivision founded!

title: Template:Elasticity/Topic news


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Template:Elasticity/Intro

Elasticity theory is the most well developed and popular topic in solid mechanics. The theory improves over previous ad hoc (strength of materials) approaches for determining the stresses and deformations of elastic materials. An elastic material is one which, after loading and subsequent removal of the load returns to its original shape (if given sufficient time).

There are several theories of elastic materials:
 * linear elasticity - valid only for infinitesimal strains
 * nonlinear elasticity - valid for a all strains
 * hypoelasticity - elasticity with dissipation
 * hyperelasticity - perfect elasticity
 * viscoelasticity - rate dependent elasticity

Most of these behaviors fall into a class of materials called simple materials.

title: Template:Elasticity/Intro


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Template:Elasticity/Participants

The histories of Wikiversity pages indicate who the active participants are. If you are an active participant in this subdivision, you can list your name here (this can help small subdivisions grow and the participants communicate better; for large subdivisions a list of active participants is not needed).

title: Template:Elasticity/Participants


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Portal:Elasticity/Learning projects

 * Introduction to Elasticity

title: Portal:Elasticity/Learning projects


text: Template:Film School:Moods:Pop Quiz:Answers


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Portal:Elasticity/Subtopics

Engineering elasticity. Advanced elasticity. Anisotropic elasticity . Theory of composites. Plates and shells. Micromechanics. Fracture mechanics. Biomechanics

title: Portal:Elasticity/Subtopics


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Portal:Elasticity/Featured learning resource

 * Introduction to Elasticity
 * Micromechanics of composites


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Portal:Elasticity/Discussion forum

 * Please discuss any changes to the learning projects here.

title: Portal:Elasticity/Discussion forum


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Template:Elasticity/Elasticity External links

iMechanica


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Template:Elasticity/Quotes

"After thousands of stretches, a pattern emerged. The molecule exhibited linear elasticity--a property that had never been seen in any other protein.” - Piotr Marszalek

title: Template:Elasticity/Quotes


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Template:Elasticity/Things you can do

 * Add material on nonlinear elasticity

title: Template:Elasticity/Things you can do


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Template:Elasticity/Research projects

To appear ...

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Template:Elasticity/WikiProjects

 * Introduction to Elasticity
 * Advanced elasticity
 * Micromechanics of composites

title: Template:Elasticity/WikiProjects


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File:Film school Gary Cline icon.jpg

Example for the WikiU Film School Lesson on Lesson:3D Storyboard

title: File:Film school Gary Cline icon.jpg


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BCP/Dianthus armeria/Logs

2007 Logs

 * --SB_Johnny | talk 09:52, 9 June 2007 (UTC) (photographic)
 * --SB_Johnny | talk 10:54, 5 July 2007 (UTC)
 * --Jomegat 17:01, 1 August 2007 (UTC)
 * --SB_Johnny | talk 12:01, 19 September 2007 (UTC)
 * --SB_Johnny | talk 14:27, 13 October 2007 (UTC)

2008 Logs

 * --SB_Johnny | talk 18:33, 21 June 2008 (UTC)
 * --Jomegat 21:21, 21 June 2008 (UTC)
 * --Jomegat 04:28, 30 June 2008 (UTC)
 * --Jomegat 23:39, 2 July 2008 (UTC)
 * --Jomegat 12:45, 8 July 2008 (UTC)
 * --SB_Johnny | talk 09:53, 14 July 2008 (UTC)
 * Jomegat 13:20, 18 July 2008 (UTC)

2009 Logs

 * (no archived logs yet)

Recent Logs

 * Jomegat 00:52, 22 July 2009 (UTC)

title: BCP/Dianthus armeria/Logs


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BCP/Verbascum blattaria/Logs

2007 Logs

 * --SB_Johnny | talk 10:00, 11 June 2007 (UTC)
 * --SB_Johnny | talk 10:57, 4 July 2007 (UTC)
 * Hazel 17:03, 3 August 2007 (UTC)
 * --SB_Johnny | PA! 10:29, 5 August 2007 (UTC)
 * --SB_Johnny | talk 12:28, 14 October 2007 (UTC) (sporadic, but on a few different plants)

2008 Logs

 * (no archived logs yet)

Recent Logs

 * --SB_Johnny | talk 11:34, 8 June 2008 (UTC)
 * --SB_Johnny | talk 09:42, 16 June 2008 (UTC)
 * --SB_Johnny | talk 09:00, 14 July 2008 (UTC)

title: BCP/Verbascum blattaria/Logs


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User talk:Brianmc

My home wiki is Wikinews, please use my talk page there for a prompt response.

Test Quiz extension
{Who this week withdrew their troops from Iraq? - Norway + Denmark - Poland - United Kingdom
 * type=""}

{What percentage of Wikimania 2007 attendees are Taiwanese? - 35% - 45% - 50% + 55%
 * type=""}

Wikiversity School of Journalism Progressing Slowly But Surely
The Wikiversity School of Journalism is progressing slowly but surely, although this Participant  has not had much time for either Wikinews or The School of Journalism.

If you are ready to Participate in The School of Journalism, just be bold, jump right in and do so. Your Wikinews experience should serve you well at The School of Journalism.

Best regards,

(s) Dennis L. Kent  (talk), Date: 2007-08-10 (August 10, 2007) Time: 0933 UTC

title: User talk:Brianmc


text:
BCP/Vicia cracca

Vicia cracca

title: BCP/Vicia cracca


text:
BCP/Vicia cracca/Logs

2007 Logs

 * --Jomegat 17:42, 1 August 2007 (UTC)
 * --Jomegat 19:10, 14 August 2007 (UTC)
 * --Jomegat 16:12, 12 September 2007 (UTC)
 * --Jomegat 17:07, 17 September 2007 (UTC)
 * --Jomegat 23:10, 21 September 2007 (UTC)
 * --Jomegat 17:08, 3 October 2007 (UTC)
 * --Jomegat 04:19, 10 October 2007 (UTC)
 * --Jomegat 16:42, 18 October 2007 (UTC)
 * --Jomegat 22:08, 25 October 2007 (UTC)
 * --Jomegat 00:34, 28 October 2007 (UTC)
 * --Jomegat 23:16, 30 October 2007 (UTC)
 * --Jomegat 22:48, 1 November 2007 (UTC)
 * --Jomegat 22:59, 5 November 2007 (UTC)

2008 Logs

 * Jomegat 19:46, 28 May 2008 (UTC)
 * Jomegat 00:37, 3 June 2008 (UTC)
 * Jomegat 23:57, 9 June 2008 (UTC)
 * Jomegat 23:11, 15 June 2008 (UTC)
 * Jomegat 22:52, 1 July 2008 (UTC)
 * Jomegat 18:48, 12 July 2008 (UTC)
 * Jomegat 01:16, 26 July 2008 (UTC)
 * Jomegat 00:37, 2 August 2008 (UTC)
 * Jomegat 20:36, 9 August 2008 (UTC)
 * Jomegat 01:13, 17 August 2008 (UTC)
 * Jomegat 20 August, 2008
 * Jomegat 23:36, 4 September 2008 (UTC)
 * Jomegat 17:54, 4 November 2008 (UTC)

2009 Logs

 * Jomegat 22:42, 14 May 2009 (UTC)
 * Jomegat 00:14, 21 May 2009 (UTC)
 * Jomegat 02:00, 4 June 2009 (UTC)
 * Jomegat 00:27, 9 June 2009 (UTC)
 * Jomegat 00:24, 15 July 2009 (UTC)
 * Jomegat 23:43, 18 July 2009 (UTC)
 * Jomegat 15:48, 26 July 2009 (UTC)
 * Jomegat 03:48, 2 August 2009 (UTC)

Recent Logs

 * Jomegat 14:42, 21 May 2010 (UTC)

title: BCP/Vicia cracca/Logs


text:
BCP/Solidago canadensis

Solidago canadensis

title: BCP/Solidago canadensis


text:
BCP/Solidago canadensis/Logs

2007 Logs

 * --Jomegat 01:06, 30 July 2007 (UTC)
 * --SB_Johnny | PA! 10:39, 5 August 2007 (UTC)
 * --Jomegat 19:09, 14 August 2007 (UTC)
 * --Luai lashire 17:47, 15 August 2007 (UTC)
 * --Jomegat 04:19, 23 August 2007 (UTC)
 * --Jomegat 12:42, 28 August 2007 (UTC)
 * --Luai lashire 14:12, 9 September 2007 (UTC)
 * --SB_Johnny | talk 10:34, 10 September 2007 (UTC)
 * --Jomegat 12:46, 10 September 2007 (UTC)
 * --Luai lashire 17:59, 15 September 2007 (UTC)
 * --Jomegat 23:08, 21 September 2007 (UTC)
 * --Luai lashire 18:30, 22 September 2007 (UTC)
 * --Jomegat 17:20, 25 September 2007 (UTC)
 * --Jomegat 03:02, 2 October 2007 (UTC)
 * --SB_Johnny | talk 13:16, 9 October 2007 (UTC)
 * --Jomegat 04:21, 10 October 2007 (UTC)
 * --Luai lashire 01:34, 14 October 2007 (UTC)
 * --Jomegat 18:16, 15 October 2007 (UTC)
 * SB Johnny-LM 09:32, 16 October 2007 (UTC)
 * --Jomegat 16:42, 18 October 2007 (UTC)
 * --Jomegat 00:35, 28 October 2007 (UTC)
 * --Jomegat 23:15, 30 October 2007 (UTC)
 * --Jomegat 23:01, 1 November 2007 (UTC)
 * --Jomegat 22:59, 5 November 2007 (UTC)

2008 Logs

 * Jomegat 04:26, 17 July 2008 (UTC)
 * --SB_Johnny | talk 10:56, 29 July 2008 (UTC) (about a week)
 * Jomegat 12:20, 29 July 2008 (UTC)
 * Jomegat 00:58, 2 August 2008 (UTC)
 * Jomegat 18:01, 9 August 2008 (UTC)
 * Jomegat 01:14, 17 August 2008 (UTC)
 * Jomegat 03:14, 21 August 2008 (UTC)
 * Jomegat 00:07, 30 August 2008 (UTC)
 * Jomegat 21:33, 1 September 2008 (UTC)
 * Jomegat 21:36, 18 September 2008 (UTC)
 * Jomegat 23:34, 3 October 2008 (UTC)
 * Jomegat 16:28, 14 October 2008 (UTC)
 * Jomegat 18:12, 3 November 2008 (UTC)

2009 Logs

 * Jomegat 14 July 2009 (from photos)
 * Jomegat 23:39, 18 July 2009 (UTC)
 * Jomegat 15:49, 26 July 2009 (UTC)
 * Jomegat 03:47, 2 August 2009 (UTC)
 * Trinity507 18:45, 30 September 2009 (UTC)
 * Jomegat 02:50, 23 October 2009 (UTC)

Recent Logs

 * Jomegat 04:05, 5 July 2010 (UTC)
 * Jomegat 19:46, 24 July 2010 (UTC)
 * User:TristanDolciano 01:28, 25 Aug 2019 (UTC)

title: BCP/Solidago canadensis/Logs


text:
User:Brianmc

My home wiki is Wikinews, please use my talk page there for a prompt response.

title: User:Brianmc


text:
Bloom Clock/Keys/New Hampshire

The pages linked below are "keys" for flowers that have been logged as blooming New Hampshire during a particular month.

title: Bloom Clock/Keys/New Hampshire


text:
Bloom Clock/Keys/New Hampshire/August/All/DPL

This list is used to create a visual key.

 category=BCP/NHAMP/8 order=descending ordermethod=lastedit


text:
Bloom Clock/Keys/New Hampshire/August/All

The plants pictured below have been recorded as blooming during the month of August in New Hampshire.

8


text:
Bloom Clock/Keys/New Hampshire/July/All/DPL

This list is used to create a visual key.

 category=BCP/NHAMP/7 order=descending ordermethod=lastedit


text:
Bloom Clock/Keys/New Hampshire/July/All

The plants pictured below have been recorded as blooming during the month of July in New Hampshire.

title: Bloom Clock/Keys/New Hampshire/July/All


text: Category:Bloom Clock Keys/NHAMP

title: Category:Bloom Clock Keys/NHAMP


text:
User talk:Rayber

My name is Ray Bergmann. I live in the city of Brisbane which is on the east coast of Australia. I am interested in natural and constructed languages. My native tongue is English and I also speak German and Japanese quite well, although my writing and reading skills in Japanese need a lot of improvement.

I collect romanised vocabulary lists in the following languages:

English – en; German (Deutsch) – de; Dutch (Nederlands) – nl; Swedish (Svenska) – sv; Norwegian (Norsk) – no; Danish (Dansk) – da; skandinavisk; Yiddish (ייִדיש) – yi;

Spanish (Español) – es; French (Français) – fr; Italian (Italiano) – it; Portuguese (Português) – pt; Latin (Latina) – la; Ladino – lad;

Simple English – simple; Lingua Franca Nova ; Interlingua – ia; Novial – nov; Interlingue – ie; Ido – io; Esperanto – eo; Volapük – vo; Lojban – jbo; slovio; Siberian / Nort Russian (Сибирской) – ru-sib;

Polish (Polski) – pl; Czech (Čeština) – cs; Slovak (Slovenčina) – sk; Slovenian (Slovenščina) – sl; Bosnian (Bosanski) – bs; Croatian (Hrvatski) – hr; Serbocroatian (Српскохрватски) – sh; Serbian (Српски) – sr; Bulgarian (Български) – bg; Russian (Русский) – ru;

Greek (Ελληνικά) – el;

Sanskrit (संस्कृतम्) – sa; Pali (पालि) – pi;

Arabic (العربية) – ar; Hebrew (עברית) – he; Maltese (Malti) – mt; Assyrian (ܣܘܪܬ) – arc;

Indonesian (Bahasa Indonesia) – id; Malay (Bahasa Melayu) – ms;

Hungarian (Magyar) – hu; Finnish – fi; Estonian – et;

Tibetan – bo; Korean – ko; Japanese (日本語) – ja;

Cantonese (粵語) – zh-yue; Chinese (中文) – zh;

title: User talk:Rayber


text:
Bloom Clock/Keys/Southeastern Pennsylvania/August/Orange Flowers/DPL

This page is used to help create a graphical key.

 category=Orange flowered plants category=BCP/SEPA/8 order=descending ordermethod=lastedit


text:
Bloom Clock/Keys/Southeastern Pennsylvania/August/Orange Flowers

The plants pictured below have been recorded as blooming during the month of August in Southeastern Pennsylvania.

last updated 09:31, 18 October 2007 (UTC) with 13 plants


text:
User:Rayber

My name is Ray Bergmann. I live in the city of Brisbane which is on the east coast of Australia. I am interested in natural and constructed languages. My native tongue is English and I also speak German and Japanese quite well, although my writing and reading skills in Japanese need a lot of improvement.

I collect romanised vocabulary lists in the following languages:

English – en; German (Deutsch) – de; Dutch (Nederlands) – nl; Swedish (Svenska) – sv; Norwegian (Norsk) – no; Danish (Dansk) – da; skandinavisk; Yiddish (ייִדיש) – yi;

Spanish (Español) – es; French (Français) – fr; Italian (Italiano) – it; Portuguese (Português) – pt; Latin (Latina) – la; Ladino – lad;

Simple English – simple; Lingua Franca Nova ; Interlingua – ia; Novial – nov; Interlingue – ie; Ido – io; Esperanto – eo; Volapük – vo; Lojban – jbo; slovio; Siberian / Nort Russian (Сибирской) – ru-sib;

Polish (Polski) – pl; Czech (Čeština) – cs; Slovak (Slovenčina) – sk; Slovenian (Slovenščina) – sl; Bosnian (Bosanski) – bs; Croatian (Hrvatski) – hr; Serbocroatian (Српскохрватски) – sh; Serbian (Српски) – sr; Bulgarian (Български) – bg; Russian (Русский) – ru;

Greek (Ελληνικά) – el;

Sanskrit (संस्कृतम्) – sa; Pali (पालि) – pi;

Arabic (العربية) – ar; Hebrew (עברית) – he; Maltese (Malti) – mt; Assyrian (ܣܘܪܬ) – arc;

Indonesian (Bahasa Indonesia) – id; Malay (Bahasa Melayu) – ms;

Hungarian (Magyar) – hu; Finnish – fi; Estonian – et;

Tibetan – bo; Korean – ko; Japanese (日本語) – ja;

Cantonese (粵語) – zh-yue; Chinese (中文) – zh;

title: User:Rayber


text:
Bloom Clock/Keys/Southeastern Pennsylvania/August/Red Flowers/DPL

This page is used to help create a graphical key.

 category=Red flowered plants category=BCP/SEPA/8 order=descending ordermethod=lastedit


text:
Bloom Clock/Keys/Southeastern Pennsylvania/August/Red Flowers

The plants pictured below have been recorded as blooming during the month of August in Southeastern Pennsylvania.

last updated 09:28, 18 October 2007 (UTC) with 12 plants


text:
Elasticity/Wedge with boundary tractions

Wedge with Boundary Tractions
Suppose Then
 * The tractions on the boundary vary as $$r^n\,$$.
 * No body forces.
 * $$\text{(1)} \qquad

\varphi = r^{n+2} f(\theta) $$ To find $$f(\theta)$$ plug into $$\nabla^4{\varphi} = 0$$.
 * $$\text{(2)} \qquad

\left(\frac{d^2}{d\theta^2} + (n+2)^2\right) \left(\frac{d^2}{d\theta^2} + n^2\right) f(\theta) = 0 $$ If $$n \ne 0$$ and $$n \ne -2$$,
 * $$\text{(3)} \qquad

\varphi = r^{n+2} \left[ a_1 \cos\{(n+2)\theta\} + a_2 \cos(n\theta) + a_3 \sin\{(n+2)\theta\} + a_4 \sin(n\theta)\right] $$ The corresponding stresses and displacements can be found from the tables associated with Michell's solution. We have to take special care for the case where $$n = 0$$, i.e., the traction on the surface is constant.

Sample Homework Problem
Find the stresses and displacements for a wedge loaded in constant shear $$ S $$ on its surface.


text:
Elasticity/Williams asymptotic solution

Williams' Asymptotic Solution
''Ref: M.L. Williams, ASME J. Appl. Mech., v. 19 (1952), 526-528.''
 * Stress concentration at the notch.
 * Singularity at the sharp corner, i.e, $$\sigma_{ij}\rightarrow\infty$$.
 * William's solution involves defining the origin at the corner and expanding the stress field as an asymptotic series in powers of r.
 * If the stresses (and strains) vary with $$r^{\alpha}$$ as we approach the point $$r = 0$$, the strain energy is given by
 * $$\text{(4)} \qquad

U = \frac{1}{2}\int_0^{2\pi}\int_0^r\sigma_{ij}\varepsilon_{ij}~r~dr~d\theta = C\int_0^r r^{2a+1} dr  $$ This integral is bounded only if $$a > -1$$. Hence, singular stress fields are acceptable only if the exponent on the stress components  exceeds $$-1$$.

Stresses near the notch corner

 * Use a separated-variable series as in equation (3).
 * Each of the terms satisfies the traction-free BCs on the  surface of the notch.
 * Relax the requirement that $$n$$ in equation (3) is an integer. Let $$ n = \lambda-1$$.
 * $$\begin{align}

\text{(5)} \qquad \varphi = r^{\lambda+1} \left[\right. & a_1 \cos\{(\lambda+1)\theta\} + a_2 \cos{(\lambda-1)\theta} +\\ & a_3 \sin\{(\lambda+1)\theta\} + a_4 \sin{(\lambda-1)\theta}\left.\right] \end{align}$$ The stresses are
 * $$\begin{align}

\sigma_{rr} = r^{\lambda-1} \left[\right. & -a_1\lambda(\lambda+1)\cos\{(\lambda+1)\theta\} -a_2\lambda(\lambda+1)\cos\{(\lambda-1)\theta\} \\ & -a_3\lambda(\lambda+1)\sin\{(\lambda+1)\theta\} -a_4\lambda(\lambda+1)\sin\{(\lambda-1)\theta\}\left.\right] \text{(6)} \qquad \\ \sigma_{r\theta}= r^{\lambda-1} \left[\right. & +a_1\lambda(\lambda+1)\sin\{(\lambda+1)\theta\} +a_2\lambda(\lambda-1)\sin\{(\lambda-1)\theta\} \\ & -a_3\lambda(\lambda+1)\cos\{(\lambda+1)\theta\} -a_4\lambda(\lambda-1)\cos\{(\lambda-1)\theta\}\left.\right] \text{(7)} \qquad \\ \sigma_{\theta\theta}= r^{\lambda-1} \left[\right. & +a_1\lambda(\lambda+1)\cos\{(\lambda+1)\theta\} +a_2\lambda(\lambda+1)\cos\{(\lambda-1)\theta\} \\ & +a_3\lambda(\lambda+1)\sin\{(\lambda+1)\theta\} +a_4\lambda(\lambda+1)\sin\{(\lambda-1)\theta\}\left.\right]\text{(8)} \qquad \end{align}$$ The BCs are $$\sigma_{r\theta} = \sigma_{\theta\theta} = 0$$ at $$\theta = \alpha$$.Hence,
 * $$\begin{align}

0= r^{\lambda-1} \lambda \left[\right. & +a_1(\lambda+1)\sin\{(\lambda+1)\alpha\} +a_2(\lambda-1)\sin\{(\lambda-1)\alpha\} \\ & -a_3(\lambda+1)\cos\{(\lambda+1)\alpha\} -a_4(\lambda-1)\cos\{(\lambda-1)\alpha\}\left.\right] \text{(9)} \qquad \\ 0= r^{\lambda-1} \lambda \left[\right. & -a_1(\lambda+1)\sin\{(\lambda+1)\alpha\} -a_2(\lambda-1)\sin\{(\lambda-1)\alpha\} \\ & -a_3(\lambda+1)\cos\{(\lambda+1)\alpha\} -a_4(\lambda-1)\cos\{(\lambda-1)\alpha\}\left.\right] \text{(10)} \qquad \end{align}$$ The BCs are $$\sigma_{r\theta} = \sigma_{\theta\theta} = 0$$ at $$\theta = -\alpha$$.Hence,
 * $$\begin{align}

0= r^{\lambda-1} \lambda \left[\right. & +a_1(\lambda+1)\cos\{(\lambda+1)\alpha\} +a_2(\lambda+1)\cos\{(\lambda-1)\alpha\} \\ & +a_3(\lambda+1)\sin\{(\lambda+1)\alpha\} +a_4(\lambda+1)\sin\{(\lambda-1)\alpha\}\left.\right] \text{(11)} \qquad \\ 0= r^{\lambda-1} \lambda \left[\right. & +a_1(\lambda+1)\cos\{(\lambda+1)\alpha\} +a_2(\lambda+1)\cos\{(\lambda-1)\alpha\} \\ & -a_3(\lambda+1)\sin\{(\lambda+1)\alpha\} -a_4(\lambda+1)\sin\{(\lambda-1)\alpha\}\left.\right]\text{(12)} \qquad \end{align}$$ The above equations will have non-trivial solutions only for certain eigenvalues of $$\lambda$$, one of which is $$\lambda = 0$$. Using the symmetries of the equations, we can partition the coefficient matrix.

Eigenvalues of λ
Adding equations (9) and (10),
 * $$\text{(13)} \qquad

a_3(\lambda+1)\cos\{(\lambda+1)\alpha\} + a_4(\lambda-1)\cos\{(\lambda-1)\alpha\} = 0 $$ Subtracting equation (10) from (9),
 * $$\text{(14)} \qquad

a_1(\lambda+1)\sin\{(\lambda+1)\alpha\} + a_2(\lambda-1)\sin\{(\lambda-1)\alpha\} = 0 $$ Adding equations (11) and (12),
 * $$\text{(15)} \qquad

a_1(\lambda+1)\cos\{(\lambda+1)\alpha\} + a_2(\lambda+1)\cos\{(\lambda-1)\alpha\} = 0 $$ Subtracting equation (12) from (11),
 * $$\text{(16)} \qquad

a_3(\lambda+1)\sin\{(\lambda+1)\alpha\} + a_4(\lambda+1)\sin\{(\lambda-1)\alpha\} = 0 $$ Therefore, the two independent sets of equations are
 * $$\text{(17)} \qquad

\begin{bmatrix} (\lambda+1)\sin\{(\lambda+1)\alpha\} & (\lambda-1)\sin\{(\lambda-1)\alpha\} \\ (\lambda+1)\cos\{(\lambda+1)\alpha\} & (\lambda+1)\cos\{(\lambda-1)\alpha\} \end{bmatrix} \begin{bmatrix} a_1 \\ a_2 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} $$ and
 * $$\text{(18)} \qquad

\begin{bmatrix} (\lambda+1)\cos\{(\lambda+1)\alpha\} & (\lambda-1)\cos\{(\lambda-1)\alpha\} \\ (\lambda+1)\sin\{(\lambda+1)\alpha\} & (\lambda+1)\sin\{(\lambda-1)\alpha\} \end{bmatrix} \begin{bmatrix} a_3 \\ a_4 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix} $$ Equations (17) have a non-trivial solution only if
 * $$\text{(19)} \qquad

\lambda\sin(2\alpha) + \sin(2\lambda\alpha) = 0 $$ Equations (18) have a non-trivial solution only if
 * $$\text{(20)} \qquad

\lambda\sin(2\alpha) - \sin(2\lambda\alpha) = 0 $$
 * From equation (4), acceptable singular stress fields must have $$\lambda > 0$$.Hence, $$\lambda = 0$$ is not acceptable.
 * The term with the smallest eigenvalue of $$\lambda$$ dominates the  solution.  Hence, this eigenvalue is what we seek.
 * $$\lambda = 1$$ leads to $$\varphi = a_4 \sin(0)$$. Unacceptable.
 * We can find the eigenvalues for general wedge angles using graphical methods.

Special case : α = π = 180°
In this case, the wedge becomes a crack.In this case,
 * $$\text{(21)} \qquad

\lambda = \frac{1}{2}, 1, \frac{3}{2}, $$ The lowest eigenvalue is $$1/2$$. If we use, this value in equation (17), then the two equations will not be linearly independent and we can express them as one equation with the substitutions
 * $$\text{(22)} \qquad

a_1 = \frac{A}{2}\sin\left(\frac{\alpha}{2}\right) ~; a_2 = -\frac{3A}{2}\sin\left(\frac{3\alpha}{2}\right) $$ where $$A$$ is a constant. The singular stress field at the crack tip is then
 * $$\begin{align}

\sigma_{rr} & = \frac{K_I}{\sqrt{2\pi r}} \left[\frac{5}{4}\cos\left(\frac{\theta}{2}\right) - \frac{1}{4}\cos\left(\frac{3\theta}{2}\right)\right] \text{(23)} \qquad \\ \sigma_{r\theta} & = \frac{K_I}{\sqrt{2\pi r}} \left[\frac{3}{4}\cos\left(\frac{\theta}{2}\right) + \frac{1}{4}\cos\left(\frac{3\theta}{2}\right)\right] \text{(24)} \qquad \\ \sigma_{\theta\theta} & = \frac{K_I}{\sqrt{2\pi r}} \left[\frac{1}{4}\sin\left(\frac{\theta}{2}\right) + \frac{1}{4}\sin\left(\frac{3\theta}{2}\right)\right] \text{(25)} \qquad \end{align}$$ where, $$K_I$$ is the { Mode I Stress Intensity Factor.}
 * $$\text{(26)} \qquad

K_I = 3A\sqrt{\frac{\pi}{2}} $$ If we use equations (18) we can get the stresses due to a mode II loading.
 * $$\begin{align}

\sigma_{rr} & = \frac{K_{II}}{\sqrt{2\pi r}} \left[-\frac{5}{4}\sin\left(\frac{\theta}{2}\right) + \frac{3}{4}\sin\left(\frac{3\theta}{2}\right)\right] \text{(27)} \qquad \\ \sigma_{r\theta} & = \frac{K_{II}}{\sqrt{2\pi r}} \left[-\frac{3}{4}\sin\left(\frac{\theta}{2}\right) - \frac{3}{4}\sin\left(\frac{3\theta}{2}\right)\right] \text{(28)} \qquad \\ \sigma_{\theta\theta} & = \frac{K_{II}}{\sqrt{2\pi r}} \left[\frac{1}{4}\cos\left(\frac{\theta}{2}\right) + \frac{3}{4}\sin\left(\frac{3\theta}{2}\right)\right] \text{(29)} \qquad \end{align}$$

title: Elasticity/Williams asymptotic solution


text:
Elasticity/Axially loaded wedge

Axially Loaded Wedge
The BCs at $$\theta = \pm \beta$$ are
 * $$\text{(30)} \qquad

t_r = t_{\theta} = 0 ~; \widehat{\mathbf{n}}{} = \pm \widehat{\mathbf{e}}{\theta} \Rightarrow \sigma_{r\theta} = \sigma_{\theta\theta} = 0 $$

What about the concentrated force BC?

 * What is $$\widehat{\mathbf{n}}{}$$ at the vertex ?
 * The traction is infinite since the force is applied on zero area. Consider equilibrium of a portion of the wedge.

At $$r = a$$, the BCs are
 * $$\text{(31)} \qquad

\widehat{\mathbf{n}}{} = \widehat{\mathbf{e}}{r} \Rightarrow \sigma_{r\theta} = t_r ~; \sigma_{\theta\theta} = t_{\theta} $$ For equilibrium, $$\sum F_1 = \sum F_2 = \sum M_3 = 0$$. Therefore,
 * $$\begin{align}

P_1 + \int_{-\beta}^{\beta} \left[ \sigma_{rr}(a,\theta)\cos\theta - \sigma_{r\theta}(a,\theta)\sin\theta\right] a~d\theta = 0 \text{(32)} \qquad \\ \int_{-\beta}^{\beta} \left[ \sigma_{rr}(a,\theta)\sin\theta + \sigma_{r\theta}(a,\theta)\cos\theta\right] a~d\theta = 0 \text{(33)} \qquad \\ \int_{-\beta}^{\beta} \left[ a \sigma_{r\theta}(a,\theta) \right] a~d\theta = 0 \text{(34)} \qquad \end{align}$$ These constraint conditions are equivalent to the concentrated force BC.

Solution Procedure
Assume that $$\sigma_{r\theta}(r,\theta) = 0$$. This satisfies the traction BCs on $$\theta = \pm\beta$$ and equation (34). Therefore,
 * $$\text{(35)} \qquad

\sigma_{r\theta} = \frac{\partial }{\partial} {}{r}\left(\frac{1}{r}\frac{\partial }{\partial} {\varphi}{\theta}\right) = 0 \Rightarrow \varphi = r\eta(\theta) + \zeta(r) $$ Hence,
 * $$\text{(36)} \qquad

\sigma_{\theta\theta} = \frac{\partial^2 }{\partial \varphi \partial r} = \zeta^{''}(r) $$ That means $$\sigma_{\theta\theta}$$ is independent of $$\theta$$. Therefore, in order to satisfy the BCs, $$\sigma_{\theta\theta} = 0$$, i.e.,
 * $$\text{(37)} \qquad

\zeta(r) = C_1 r + C_2 \Rightarrow \varphi = r\eta(\theta) + C_1 r = r[\eta(\theta)+C_1] = r\xi(\theta) $$ Checking for compatibility, $$\nabla^4{\varphi} = 0$$, we get
 * $$\text{(38)} \qquad

\xi^{(IV)}(\theta) + 2\xi^{''}(\theta) + \xi(\theta) = 0 $$ The general solution is
 * $$\text{(39)} \qquad

\xi(\theta) = A\sin\theta + B\cos\theta + C\theta\sin\theta + D\theta\cos\theta $$ Therefore,
 * $$\text{(40)} \qquad

{ \varphi = r\left[A\sin\theta + B\cos\theta + C\theta\sin\theta + D\theta\cos\theta\right] } $$ The only non-zero stress is $$\sigma_{rr}$$.
 * $$\text{(41)} \qquad

\sigma_{rr} = \frac{1}{r}\left[2C\cos\theta - 2D\sin\theta\right] $$ Plugging into equation (33), we get
 * $$\text{(42)} \qquad

-D\left[2\beta - \sin(2\beta)\right] = 0 \Rightarrow D = 0 $$ Hence,
 * $$\text{(43)} \qquad

\sigma_{rr} = \frac{2C}{r}\cos\theta $$ Plugging into equation (32), we get
 * $$\text{(44)} \qquad

-P = C\left[2\beta + \sin(2\beta)\right] \Rightarrow C = \frac{-P}{2\beta+\sin(2\beta)} $$ Therefore,
 * $$\text{(45)} \qquad

{ \varphi = Cr\theta\sin\theta = \frac{-P r\theta\sin\theta}{2\beta+\sin(2\beta)}} $$ The stress state is
 * $$\text{(46)} \qquad

{ \sigma_{rr} = -\frac{2P\cos\theta}{r[2\beta+\sin(2\beta)]} ~; \sigma_{r\theta} = 0 ~; \sigma_{\theta\theta} = 0 } $$

Special Case

 * $$\beta = \pi/2$$

A concentrated point load acting on a half plane.

 * $$\text{(47)} \qquad

{ \sigma_{rr} = -\frac{2P\cos\theta}{\pi r} ~; \sigma_{r\theta} = 0 ~; \sigma_{\theta\theta} = 0 } $$

Displacements

 * $$\begin{align}

2\mu u_r & = -\frac{\partial }{\partial} {\varphi}{r} + \alpha r \frac{\partial }{\partial} {\psi}{\theta} \\ 2\mu u_{\theta} & = -\frac{1}{r}\frac{\partial }{\partial} {\varphi}{\theta} + \alpha r^2 \frac{\partial }{\partial} {\psi}{r} \end{align}$$ where
 * $$\begin{align}

\nabla^2{\psi} = 0 \\ \frac{\partial }{\partial} {}{r}\left(r\frac{\partial }{\partial} {\psi}{\theta}\right) = \nabla^2{\varphi} \end{align}$$ Plug in $$\varphi = Cr\theta\sin\theta$$,
 * $$\begin{align}

& \frac{\partial }{\partial} {}{r}\left(r\frac{\partial }{\partial} {\psi}{\theta}\right)= \nabla^2{\varphi} \\ \Rightarrow & \frac{\partial }{\partial} {}{r}\left(r\frac{\partial }{\partial} {\psi}{\theta}\right)= \frac{2C}{r}\cos\theta \\ \Rightarrow & r\frac{\partial }{\partial} {\psi}{\theta}= 2C \ln r\cos\theta + A(\theta) \\ \Rightarrow & \frac{\partial }{\partial} {\psi}{\theta}= 2C \frac{\ln r}{r} \cos\theta + \frac{A(\theta)}{r} \\ \Rightarrow & \psi= 2C \frac{\ln r}{r} \sin\theta + \frac{\eta(\theta)}{r} + \xi{r} \end{align}$$ Plug $$\psi$$ into $$\nabla^2{\psi} = 0$$,
 * $$\begin{align}

& \frac{1}{r^3}\eta^{''}(\theta) + \frac{1}{r^3}\eta(\theta) + \xi^{''}(r) + \frac{1}{r}\xi^{'}(r) -\frac{4C}{r^3}\sin\theta= 0 \\ \Rightarrow & \eta^{}(\theta) + \eta(\theta) + r^3\xi^{}(r) + r^2\xi^{'}(r) - 4C\sin\theta= 0 \end{align}$$ Hence,
 * $$\begin{align}

\eta^{''}(\theta) + \eta(\theta) - 4C\sin\theta & = b\\ r^3\xi^{''}(r) + r^2\xi^{'}(r) & = -\frac{b}{r^3} \end{align}$$ Solving,
 * $$\begin{align}

\eta(\theta) & = -2C\theta\cos\theta + d\cos\theta + e\sin\theta + b\\ \xi^{'}(r) & = \frac{f}{r} + \frac{b}{r^2} \end{align}$$ Therefore,
 * $$\begin{align}

2\mu u_r & = 2\alpha C \ln r \cos\theta + (2\alpha-1)C\theta\sin\theta + \alpha(e - 2C)\cos\theta - \alpha d\sin\theta \\ 2\mu u_{\theta} & = -2\alpha C \ln r \sin\theta + (2\alpha-1)C\sin\theta + (2\alpha - 1)C\theta\cos\theta - \alpha d\cos\theta -\alpha e \sin\theta + \alpha f r \end{align}$$ To fix the rigid body motion, we set $$u_{\theta} = 0$$ when $$\theta = 0$$, and set $$u_r = 0$$ when $$\theta = 0$$ and $$r = L$$.Then,
 * $$\begin{align}

u_r & = \frac{\alpha C}{\mu}\ln\left(\frac{r}{L}\right)\cos\theta + \frac{(2\alpha-1)C}{2\mu}\theta\sin\theta \\ u_{\theta} & = -\frac{\alpha C}{\mu}\ln\left(\frac{r}{L}\right)\sin\theta + \frac{(2\alpha-1)C}{2\mu}\theta\cos\theta - \frac{C}{2\mu}\sin\theta \end{align}$$ The displacements are singular at $$r = 0$$ and $$r = \infty$$. At $$\theta = 0$$,
 * $$\begin{align}

u_r & = \frac{\alpha C}{\mu}\ln\left(\frac{r}{L}\right)\\ u_{\theta} & = 0 \end{align}$$ Is the small strain assumption satisfied ?

title: Elasticity/Axially loaded wedge


text:
Elasticity/Flamant solution

The Flamant Solution
From Michell's solution, pick terms containing $$1/r$$ in the stresses. Then,
 * This problem is also self-similar (no inherent length scale).
 * All quantities can be expressed in the separated-variable form  $$\sigma = f(r)g(\theta)$$.
 * The stresses vary as $$(1/r)$$ (the area of action of the force  decreases with increasing $$r$$).  How about a conical wedge ?

\varphi = C_1 r \theta\sin\theta + C_2 r\ln r \cos\theta + C_3 r \theta\cos\theta + C_4 r\ln r \sin\theta $$ Therefore, from Tables,
 * $$\begin{align}

\sigma_{rr} & = C_1\left(\frac{2\cos\theta}{r}\right) + C_2\left(\frac{\cos\theta}{r}\right) + C_3\left(\frac{2\sin\theta}{r}\right) + C_4\left(\frac{\sin\theta}{r}\right) \\ \sigma_{r\theta} & = C_2\left(\frac{\sin\theta}{r}\right) + C_4\left(\frac{-\cos\theta}{r}\right) \\ \sigma_{\theta\theta} & = C_2\left(\frac{\cos\theta}{r}\right) + C_4\left(\frac{\sin\theta}{r}\right) \end{align}$$ From traction BCs, $$C_2 = C_4 = 0$$. From equilibrium,
 * $$\begin{align}

F_1 + 2\int_{\alpha}^{\beta} \left(\frac{C_1\cos\theta - C_3\sin\theta}{a}\right)a\cos\theta d\theta & = 0 \\ F_2 + 2\int_{\alpha}^{\beta} \left(\frac{C_1\cos\theta - C_3\sin\theta}{a}\right)a\sin\theta d\theta & = 0 \end{align}$$ After algebra,

\sigma_{rr} = \frac{2C_1\cos\theta}{r} + \frac{2C_3\sin\theta}{r} ~; \sigma_{r\theta} = 0 ~; \sigma{\theta\theta} = 0 $$

Special Case : α = -π, β = 0

C_1 = - \frac{F_1}{\pi} ~; C_2 = \frac{F_2}{\pi} $$ The displacements are
 * $$\begin{align}

u_1 & = - \frac{F_1(\kappa+1)\ln|x_1|}{4\pi\mu} + \frac{F_2(\kappa+1)\text{sign}(x_1)}{8\mu} \\ u_2 & = - \frac{F_2(\kappa+1)\ln|x_1|}{4\pi\mu} - \frac{F_1(\kappa+1)\text{sign}(x_1)}{8\mu} \end{align}$$ where
 * $$\begin{align}

\kappa = 3 - 4\nu & & \text{plane strain} \\ \kappa = \frac{3 - \nu}{1+\nu} & & \text{plane stress} \\ \end{align}$$ and

\text{sign}(x) = \begin{cases} +1 & x > 0 \\ -1 & x < 0 \end{cases} $$

title: Elasticity/Flamant solution


text:
User:Vivelefrat

fr:Utilisateur:Vivelefrat

fr:Utilisateur:Vivelefrat it:Utente:Vivelefrat

title: User:Vivelefrat


text:
Russian/Introduction

Alphabet
The Russian alphabet has 33 letters, out of which 10 are vowels and 21 are consonants. Two of the letters (Ь and Ъ) are used for changing sound of the preceding consonant.

Russian Pronunciation 1, The Basic 33 Character Alphabet

You have reached the end of the Cyrillic alphabet. Now go and practice in Lesson 1!

Phonetics
The greatest problem in Russian phonetics for English speakers is Palatalization. Almost all the consonants in the Russian language come in two flavors: non-palatalized (called "hard") and palatalized (called "soft"). The quality of a consonant mostly depends on the following vowel and is reflected in writing. If the following letter is И, Е, Ю, Я or Ё thеn the consonant is soft, palatalized, that means that while pronouncing it you have to raise the middle part of your tongue. If the following consonant letter is А, О, У, Ы, Э then the consonant is hard, non-palatalized. Palatalization of the final consonant as well as sometimes that of the consonant before another consonant is marked with the letter Ь. Some consonants are always hard ( Ж, Ш, Ц ) and some are always soft ( Щ, Ч ) whatever letter follows. E.g. the last sound in мышь (mouse) is the same as the first one in шàпка (warm hat) and is hard.

Articles
Articles are not used in Russian. Articles are a part of speech that indicates, specifies and limits a noun (a, an, or the in English). In some languages the article may appear as an ending (e.g. definite article in Swedish), in other languages like Russian articles are not used.

Nouns
As with many other languages, Russian is devised of several "cases" for their nouns. These cases reflect the context of how the noun is used and aids in understanding. The cases are: Nominative, Genitive, Prepositional, Accusative, Dative, and Instrumental. Furthermore, nouns also have an inherent Gender and Palatalization (or hard vs soft).

Nominative Case
The Nominative case is used for the subject of a sentence. For example, "The cat is here!" = "Кошка тут!" Note: Russian lacks the present tense of the verb "to be." If you wish to declare something, you would say "Это кошка!" or "This is a cat!"

As you can see in the chart to the right, Masculine Nominative nouns tend to end in consonants, while Feminine Nominative nouns end in "a" or "я," and Neuter Nominative nouns end in "е" or "о."

Another use for the Nominative case is labeling. For example, if you wished to say that you are a student, you would simply say "Я студент/студентка" (male or female, respectively). Again, there is no Russian for the present tense of "to be" so "is" is dropped. This works the same for adjectives, as you can label something with the Nominative case: "Студент - умный," or "the student is smart."

More examples of the nominative case:
 * Джон учит русский язык. = John studies Russian.
 * John is the nominative because he is performing the action.
 * Кошка пьет молоко. = The (female) cat drinks milk.
 * The cat is the nominative because it is performing the action.
 * Апельсин - фрукт = Orange is a fruit.
 * Nominative case is roughly the equivalent of the 'to be' structure in English.
 * Она - учитель = She is a teacher. (NB:Russians are very particular about usage of teacher and professor).
 * Russians use a dash to indicate the verb 'to be'.
 * Стул отремонтирован. = The (♂)chair is repaired.
 * The chair is the nominative because it was acted upon. In this instance the acting agent is unknown. This is the passive voice in English.

Accusative Case
The Accusative case is used with direct objects and certain prepositions. For example, "I love you" = "Я люблю тебя," where "тебя" is the accusative form of "you." Furthermore there exists a distinction between "animate" and "inanimate" nouns. Animate nouns include people and animals. Inanimate nouns include mostly everything else. Masculine inanimate nouns do not change from the Nominative case. Masculine animate nouns take on the GENITIVE case when used as a direct object or with a preposition. Feminine and neuter nouns do not make this distinction (for now). To illustrate this:

The prepositions that use the Accusative case are в, на. Using these prepositions with the accusative case of the noun mean "direction towards a place." This is opposed to using в, на with the prepositional case, which means location. That is for later, however.

Numbers
In Russian, using numbers above twenty is similar to English: you take the base 20 ("двадцать") and then add the single number ("один"). For example: 21 = двадцать один, 55 = пятьдесят пять, 126 = сто двадцать шесть.

Note that these numbers are cardinal numbers, which are not the same as ordinal numbers, i.e. "first" or "second." Cardinal numbers are treated as adjectives in Russian and follow a slightly different pattern.

How to Count from 1 to 10 in Russian

Grammar of Numbers
When using Russian numbers to quantify an amount, i.e. four dogs, the case of the noun is dependent on the preceding number. If you have one object, you retain the Nominative singular case. If you have two to four of an object, you use the Genitive singular case. If you have five or more, then you use the Genitive plural case. For zero, you use the Genitive plural case. This is not to be confused with the phrase "Нет," which translates to "There is no" or "No such * exists," in which the case is usually the genitive singular.

Further practice and how to say and ask time in Lesson 2.

Pronouns
There are several groups of pronouns in Russian language. The first is Personal Pronoun. A pronoun functions in the same way as the noun that it replaces.

Adjective
Adjectives in Russian are declined based on the case, gender and plurality of the noun that they modify. In Russian, genders are broken down into: Masculine, Feminine and Neuter. There are six cases: Nominative, Accusative, Genitive, Dative, Instrumental, and Prepositional. For plurality, there is singular and plural. This can get very complicated, however.

Verbs
The Russian language has 3 basic tenses: present, past and future.

The Present Tense

First Conjunction

Let's take the Russian verb in its infinitive form "работать" as an example. Put "ть" aside and add the additions which are shown below.

 * работать  -   To work.
 * Я работаю  -   I work
 * Ты работаешь  -   You work (singular and informal)
 * Он, Она, Оно работает	 -   He, She, It works.
 * Мы работаем	 -   We work
 * Вы работаете	 -   You work (plural and formal)
 * Они работают	 -   They work.

Second Conjunction

Some verbs which end in "ить" have got some differences. For example;

д becomes ж, з becomes ж, c becomes ш, ct becomes щ, т becomes ч, and for б, в, м, п, ф add the letter л. Remember, this only applies to the first-person singular (Я).

•	говорить	  = To speak.

•	Я говорю	 = I speak.

• Ты говоришь   =   You speak (singular).

• Он, Она, Оно говорит    =   He, She, It Speaks.

• Мы говорим    =   We speak.

• Вы говорите    =   You speak (plural).

• Они говорят    =   They speak.

Saying "What" in Russian
Что is used as an interrogative pronoun. Likewise, if you see an object you have never seen before, and have no idea what it is, then you ask the question of "что это?" which means "What's this?" or "What's that?"

 * Что ты делаешь? =What are you doing?
 * Что мы хотим есть? =What do we want to eat?
 * Что мне купить в магазине? =What should I buy at the store?

Что такое is used when the object has been named or introduced, but you want more information on it. For instance: Что это? Это машина! Что такое машина? translates to: What's this? It's a machine! What's a machine?

 * Что такое компьютер? (What is a computer?)
 * Что такое водка? (What is vodka? [although this phrase is rarely spoken in Russian])
 * Что такое Южная Осетия? (What is South Ossetia?)

If you wish to ask what type or variant something is, like in the sentence: "What type of computer is this?" then you use the adjective какой which declines depending upon its case. Note, it is an irregular adjective.

 * Какая эта водка? =What kind of vodka is this?
 * Какой это компьютер? =What kind of computer?
 * Какое это слово? =What kind of word is that?

If you ask what a living creature, like a pet or a family member, is, you use the word кто (who), and apply it to the question: Кто это?

Saying "For" in Russian
Russian has many words that translate to the English "for". Here are some examples.

For recipient use of the word "for" such as "I have a present for you.", one uses the Dative cases.

 * Это всё мне? (Is this for me?)
 * Я принёс новые игры Ивану (I brought new games for Ivan.)
 * Когда я куплю новую шапку папе? (When will I buy a new cap for father?)

For expressing "for the benefit of", such as "Anna reads better than Ivan, therefore Anna reads for him.", one uses для + Genitive.

 * Мне нужна помощь для школы. (I need help for school.)
 * Я сделала это для него. (I did it for him.)
 * Ты можешь прочитать газету для твоей бабушки? (Can you read the newspaper for your grandmother?)

For expressing "exchange" or "replacement", one uses за + Accusative.

 * Они говорят за меня. (They spoke [instead of/for] me.)
 * Спасибо за билеты. (Thanks for the tickets.)
 * Дай мне совет за деньги! (Give me advice for money!)[often used ironically]

Duration of time (During the action) uses the Accusative case.

Duration of time (After the action) uses на + Accusative.
 * Он пил три часа (He has been drinking for three hours.)

 * Он вышел на 5 минут (He left for 5 minutes.)

To express "to get" as in "He went for some food." one uses за + Instrumental.

 * Она пошла в магазин за продуктами. (She went to the store for products/items/groceries.)
 * Он пошёл на кухню за чипсами. (He went to the kitchen for chips.)

To express "I am for taxes." as in "I support taxes", one uses за + Accusative. (против + Genitive is the opposite.)

 * Я за социализм. (I am for socialism./I support socialism.)
 * Я против социализма. (I am not for socialism./I do not support socialism.)

Last, to express "for" in the sense of a deadline or scheduled time, one uses на + Accusative.

 * Я еду в Россию на зиму. (I am going to Russia for the winter.)
 * Задание на завтра. (Homework for tomorrow.)

working
Они начали работу. =They started working.

living

 * Это моя квартира. =This is my apartment.
 * Мы живём большом доме. = We live in a big house.
 * Вы живёте в доме или в квартире? = Do you live in a house or in an apartment? (formal)

clarification

 * понимаете меня? = Do you understand me? (inf)
 * Ну, видите ль? = Well, you see eh?
 * Да чего вы скупитесь? = Yes, what do you mean?
 * Что это за люди? = Who are these people?
 * вы уж собираетесь ехать? = You really want to go.

apologies

 * Извините, [iz-vee-NEAT-ye] = Sorry,
 * Простите, [pras-TEET-ye] = Excuse me.
 * Простите, я не подумал. = I'm sorry, I was not thinking.

Danger

 * Это очень опасно! = That is dangerous!

Warning

 * Не ставь! = Don't put (it there).
 * Держи! = Hold it!
 * Срочно! = It's urgent!

accepting apologies

 * Ничего!= Never mind!, Don't mention it!, It doesn't matter!
 * Не беспокойтесь = do not worry

informal greetings and good byes

 * Как дела? [kak dee-LA] = How are you?
 * Как поживаешь? = How are you?
 * Привет! [pri-VYET] = Hello! (Informal)
 * Здравствуйте! [ZDRAST-vooy-tye] = Hello! (Formal)
 * Пока! [pa-KA] = Bye! (Informal)
 * До свидания! [da-svee-DAN-ee-ya] = Bye! (Formal)
 * Очень приятно! [O-chin Pri-YAT-na] Nice to meet you! (Formal)
 * Каково почивали? = Rested?

opposites

 * Хорошо [hara-SHO] = Fine
 * Плохо [PLO-ha] = Bad
 * Вроде нет. = No, sort of.
 * Вроде да. = Yes, sort of.

help!

 * Мне нужна помощь! [mnyeh noozh-NAH PO-mash'] = I need help. Help!
 * потому, что я тупенький. = Because I'm stupid.

introductions

 * Как тебя зовут? [kak tee-BYA za-VOOT] = What is your name?
 * Меня зовут... [mee-NYA za-VOOT] = My name is…

miscellaneous

 * Так вы полагаете? = So what do you think?
 * какое пекло! = What the hell?

informing

 * В туалете грязно. = The bathroom is dirty.
 * Окна грязные. = The windows are dirty. (note change in грязн*)
 * Пол влажный. = The floor is wet.
 * Я устал! = I am tired.
 * Важно, что я – тут = What's important is that I am here.

coming and going

 * мы уходим. = We are leaving.
 * -Я вернусь. = I'll be back.

distance phrases
далеко они? = Are they far?

quiz 1
{What does Я прибываю mean? +Я прибываю = I arrive -Я прибываю = You arrive -Я прибываю = He arrives -Я прибываю = She arrives
 * type=""}

{Pick the best meaning of: для какие причин вам это нужно? +Why do you need it? -What is is worth? -Who is asking?
 * type=""}

quiz 2 directions
{What does:Вы не подскажете, где здесь туалет?mean? +Excuse me, is there a bathroom around here? -To the left -Are you in a hurry? -To the right -Where is the entrance?
 * type=""}

{What does:налево [nah-LYE-va]mean? +To the left -Are you in a hurry? -To the right -Where is the entrance?
 * type=""}

{What does:направо [nah-PRA-va]mean? +To the right -To the left -Are you in a hurry? -Where is the entrance?
 * type=""}

{What does:вы так спешите?mean? +Are you in a hurry? -To the left -Where is the entrance? -To the right
 * type=""}

{What does:Скажите пожалуйста, где здесь банк?mean? +Excuse me, is there a bank around here? -To the left -Where is the entrance? -To the right
 * type=""}

{What does:Где здесь вход?mean? +Where is the entrance? -Excuse me, is there a bank around here? -To the left -To the right
 * type=""}

/External links/
On the subpage, place links to web sites which provide free resources that may aid in the learning of Russian. Regular Wikiversitans, participating generally here, may add or comment on links. See instructions on the subpage. Please do not add new links here without discussion. Such links may be removed without further attention.

title: Russian/Introduction


text:
Digital media workshop

The Digital Media Workshop is where you can learn about how to make digital media files such as OGG format audio and video. If you need help with a media project, you can post a request here.

Collaboration
Looking for help with a digital media project? List requests for help here: I think that by myself is nice
 * Originally at the Colloquium: Hello everybody for the english departement of fr wikiversity, I would like oral comprehension be created. So, I want some help because it's better an english person record the sound. If you are interested send me a message Thank you Vivelefrat 13:49, 5 August 2007 (UTC)

 * I'm looking for people who are interested in collaborating on astronomical image processing. The project involves me taking images with a digital camerar attached to a telescope and then working with others to convert the raw images into visually attractive pictures that can be used to illustrate astronomy lessons at wikiversity or to add to articles at wikipedia.  See my post at School talk:Graphics for more information, plus some nice samples by JWSchmidt.  --mikeu 18:06, 19 December 2007 (UTC)
 * mikeu - I am interested in using some of your astronomical images as previously discussed in games and in venture planning exercises at Lunar Boom Town. I am thinking about an exercise in the teleoperated theme park rovers of starting with your photograph of Copernicus as top level planning tool then pulling more detailed closeups from the U.S.G.S. online Planetary Data System.  Maybe mining assessments as well.  Thanks for providing useful pictures of pieces of the universe for our use! cc mikeu talk page.  Mirwin 18:12, 19 December 2007 (UTC)

 * I'd like to make simple audio files of tones for ear training. Basically, two notes played one after the other. --HappyCamper 17:16, 29 May 2011 (UTC)

Learning Resources
Wikiversity has a growing collection of learning resources about how to create and work with digital files (text, audio, video).
 * Wiki Campus Radio
 * Audacity
 * Introduction to GarageBand
 * Wikiversity Reports
 * Podcasting
 * Web 2.0 - creating internet content including digital audio and video files
 * Wikiversity the Movie
 * Filmmaking

ě==See also==
 * Topic:Internet audio and video

External resources
Video Problems: A green bar is sometimes seen on the edge of the video. This can be due to using a pixel value that is not a multiple of 16. Also, sometimes the QuickTime player does not automatically show the video (there is just a blank space on the page). The Cortado player seems to have problems keeping up with the data flow: is there an optimal bit rate?
 * Philip Greenspun illustration project - get paid to make illustrations
 * Wikipedia:Creation and usage of media files#Video
 * Help:Converting video

External resources

 * image finding tool
 * Illustration project - get paid for creation of “key illustrations” that will be used by Wikimedia wiki projects. Philip Greenspun illustration project

title: Digital media workshop


text:
BCP/Nuphar variegata/Logs

2007 Logs

 * --Jomegat 15:26, 14 July 2007 (UTC)
 * --Jomegat 16:03, 14 July 2007 (UTC)

2008 Logs

 * Jomegat 23:02, 25 May 2008 (UTC)
 * Jomegat 21:07, 21 June 2008 (UTC)
 * Jomegat 01:11, 3 July 2008 (UTC)
 * Jomegat 21:31, 12 July 2008 (UTC)
 * Jomegat 00:56, 20 July 2008 (UTC)
 * Jomegat 19:49, 26 July 2008 (UTC)
 * Jomegat 19:12, 2 August 2008 (UTC)
 * Jomegat 20:33, 9 August 2008 (UTC)
 * Jomegat 01:18, 17 August 2008 (UTC)
 * Jomegat 23:58, 23 August 2008 (UTC)
 * Jomegat 01:04, 31 August 2008 (UTC)

2009 Logs

 * Jomegat 00:29, 25 May 2009 (UTC)
 * Jomegat 02:08, 1 June 2009 (UTC)
 * Jomegat 20:33, 27 June 2009 (UTC)
 * Jomegat 03:59, 19 July 2009 (UTC)
 * Jomegat 18:39, 2 August 2009 (UTC)

Recent Logs

 * Jomegat 00:29, 30 May 2010 (UTC)
 * Jomegat 19:46, 26 June 2010 (UTC)

title: BCP/Nuphar variegata/Logs


text:
BCP/Nuphar variegata

Nuphar variegata

title: BCP/Nuphar variegata


text:
Urantia Book/Introduction

If you have come to Wikiversity to learn about the Urantia Book, you should not expect to learn about it's mysterious origin or anonymous authorship. A great deal of material has been written on those subjects that is very easily found, on Wikipedia and elsewhere. These pages on Wikiversity and meant to teach the student the lessons put forth in the Urantia Book. The book essentially teaches one to cultivate a personal relationship with our spiritual father, God, while encouraging the highest ideals of social relationships as the method of creating spiritual experiences with our mortal brothers that form the foundation of our potentially infinite existence.

Outline
The following is an abridgment of David Kantor's outline of the papers; in the introduction, he states: "There are four summary papers which may be considered separately. These are the Foreword, Paper 56, Paper 118 and Paper 196. The Foreword and Paper 56 studied together will provide significant insights, but to be productive such a study must be preceeded by a thorough reading of Papers 1 through 55."

"Paper 118 examines the theological implications of the materials developed in Parts I, II and III, and provides a conceptual foundation upon which future theological insights may be developed through a combination of rational philosophic extension and personal revelation."

"Paper 196 provides a meditative reflection on the implications of the faith of Jesus for our personal spiritual lives."

 * 1) Foreword: Orientation to some conceptual structures and relationships developed in the text.
 * 2) Papers 1-10: Primary Deity Relationships; Primary cosmological foundations of The Urantia Book's model of reality.
 * 3) Papers 11-15: Physical Cosmography; Overview of the architecture of the master universe in the present universe age; the relationship between existential and evolutionary structures and systems.
 * 4) Papers 16-31: Personality in the Universe; Introduction to the interpersonal Deity relationships which constitute the architectural structure of the personal cosmos in the present universe age.
 * 5) Papers 32-40: The Local Universe of Nebadon
 * 6) Papers 41-42: Physical Energy
 * 7) Papers 43-44: The Constellations
 * 8) Papers 45-55: The Local System
 * 9) Paper 56: A Unified Cosmology. A philosophic essay on the nature of the cosmic unity integrating the realities explored in The Urantia Papers. Paper 56 is a special Paper; Papers 1-55 build a conceptual foundation upon which the philosophic developments in Paper 56 reasonably may be made. Paper 56 also summarizes the cosmology presented in the preceeding papers. Paper 56 and the Foreword should be studied together as a review of the material in the first 55 Papers.
 * 10) Papers 57-65: The Evolutionary Process
 * 11) Papers 66-77: The Fostering of Civilization
 * 12) Papers 78-92: Evolution of Human Institutions
 * 13) Papers 93-103: Conceptual Foundations of the Fourth Epoch. Recent epochal revelations and the development of present day levels of theological understandings about the nature of Deity.
 * 14) Papers 104-118: Conceptual Foundations for the Fifth Epoch. The revelators compose the highest achievements of human theology and philosophy into a platform from which future philosophical and theological development may proceed. This involves a revealed integration of Trinitarian and Process theologies. If it is felt that the revelators had a specific purpose in ordering the Papers, the student might ask why the gap in the material in this section between that related to the Trinity and that related to the Supreme is bridged with a discussion of the Thought Adjusters and the Seraphim of Planetary Supervision.
 * 15) Papers 119-121:  The Context of Michael's Bestowal
 * 16) Papers 122-134: Jesus of Nazareth
 * 17) Papers 135-147: Beginning the Public Work. The public work of Jesus, the son of man.
 * 18) Papers 148-158: Continuing the Public Work
 * 19) Papers 159-170: Concluding the Public Work
 * 20) Papers 171-188: Arrest, Trial, and Crucifixion of Jesus
 * 21) Papers 189-193: The Post-Resurrection Jesus
 * 22) Papers 194-196: Into the Fifth Epoch

title: Urantia Book/Introduction


text:
User:Hazel~enwikiversity

I live in the Rogue Valley, in Southwestern Oregon, where I pay attention to wild plants, among other things.

title: User:Hazel~enwikiversity


text:
File:Morphs and D-forms.png

Summary
This screenshot is illustrates use of the DAZ|Studio software at DAZ Studio D-Form tool.

title: File:Morphs and D-forms.png


text:
Elasticity/Torsion of circular cylinders

About the problem:

 * Circular Cylinder.
 * Centroidal axis thru the center of each cross section (c.s.)
 * Length $$L$$, Outer radius $$c$$.
 * Applied torque $$T$$.
 * Angle of twist $$\phi$$.

Assumptions:

 * Each c.s. remains plane and undistorted.
 * Each c.s. rotates through the same angle.
 * No warping or change in shape.
 * Amount of displacement of each c.s. is proportional to  distance from end.

Find:

 * Shear strains in the cylinder ($$\gamma$$).
 * Shear stress in the cylinder ($$\tau$$).
 * Relation between torque ($$T$$) and angle of twist ($$\phi$$).
 * Relation between torque ($$T$$) and shear stress ($$\tau$$).

Solution:
If $$\gamma$$ is small, then
 * $$\text{(1)} \qquad

L\gamma = r\phi \Rightarrow {\gamma = \frac{r\phi}{L}} $$ Therefore,
 * $$\text{(2)} \qquad

\gamma_{\text{max}} = \frac{c\phi}{L} \Rightarrow \gamma = \frac{r}{c} \gamma_{\text{max}} $$ If the material is linearly elastic,
 * $$\text{(3)} \qquad

\tau = G\gamma \Rightarrow {\tau = \frac{r\phi G}{L}} $$ Therefore,
 * $$\text{(4)} \qquad

\tau_{\text{max}} = \frac{c\phi G}{L} \Rightarrow \tau = \frac{r}{c} \tau_{\text{max}} $$ The torque on each c.s. is given by
 * $$\text{(5)} \qquad

T = \int_A \tau r dA = \frac{\phi G}{L}\int_A r^2 dA = \frac{G\phi J}{L} $$ where $$J$$ is the polar moment of inertia of the c.s.
 * $$\text{(6)} \qquad

J = \begin{cases} \frac{1}{2} \pi c^4 & \text{solid circular c.s.} \\ \frac{1}{2} \pi (c_2^{~4}-c_1^{~4}) & \text{ annular circular c.s.} \end{cases} $$ Therefore,
 * $$\text{(7)} \qquad

{\tau = \frac{Tr}{J}} \Rightarrow \tau_{\text{max}} = \frac{Tc}{J} $$ and
 * $$\text{(8)} \qquad

{\phi = \frac{TL}{JG}} $$


text:
Elasticity/Torsion of noncircular cylinders

About the problem

 * Solution first found by St. Venant.
 * Tractions at the ends are statically equivalent to equal and opposite torques $$\pm \mathbf{T} = \pm T \widehat{\mathbf{e}}{3} \,$$.
 * Lateral surfaces are traction-free.

Assumptions:

 * An axis passes through the center of twist ($$x_3\,$$ axis).
 * Each c.s. projection on to the $$x_1-x_2\,$$ plane rotates,but remains undistorted.
 * The rotation of each c.s. ($$\phi\,$$) is proportional to $$x_3\,$$.

\phi = \alpha x_3 \, $$ where $$\alpha\,$$ is the twist per unit length.
 * The out-of-plane distortion (warping) is the same for each c.s. and is proportional to $$\alpha\,$$.

Find:

 * Torsional rigidity ($$T/\alpha\,$$).
 * Maximum shear stress.

Displacements

 * $$\begin{align}

u_1 & = r\cos(\phi+\theta) - r\cos\theta = x_1(\cos\phi-1)-x_2\sin\phi \\ u_2 & = r\sin(\phi+\theta) - r\sin\theta = x_1\sin\phi+x_2(\cos\phi-1)\\ u_3 & = \alpha\psi(x_1,x_2) \end{align}$$ where $$\psi(x_1,x_2)\,$$ is the warping function.

If $$\phi = \alpha x_3 << 1 \,$$ (small strain),
 * $$\text{(10)} \qquad

{ u_1 \approx -\alpha x_2 x_3 ~; u_2 \approx \alpha x_1 x_3 ~; u_3 = \alpha\psi(x_1,x_2) } $$

Strains

\varepsilon_{ij} = \frac{1}{2}\left(u_{i,j} + u_{j,i}\right) $$ Therefore,
 * $$\begin{align}

\varepsilon_{11} & = \frac{1}{2}\left(0 + 0\right) = 0 \\ \varepsilon_{22} & = \frac{1}{2}\left(0 + 0\right) = 0 \\ \varepsilon_{33} & = \frac{1}{2}\left(0 + 0\right) = 0 \\ \varepsilon_{kk} & = \varepsilon_{11} + \varepsilon_{22} + \varepsilon_{33} = 0 \\ \varepsilon_{12} & = \frac{1}{2}\left(-\alpha x_3 + \alpha x_3 \right) = 0 \\ \varepsilon_{23} & = \frac{1}{2}\left(\alpha\psi_{,2} + \alpha x_1\right) \text{(11)} \qquad \\ \varepsilon_{31} & = \frac{1}{2}\left(\alpha\psi_{,1} - \alpha x_2\right) \text{(12)} \qquad \end{align}$$

Stresses

\sigma_{ij} = 2\mu\varepsilon_{ij} + \lambda\varepsilon_{kk}\delta_{ij} $$ Therefore,
 * $$\begin{align}

\sigma_{11} & = 0 \\ \sigma_{22} & = 0 \\ \sigma_{33} & = 0 \\ \sigma_{kk} & = 0 \\ \sigma_{12} & = 0 \\ \sigma_{23} & = \mu\alpha(\psi_{,2} + x_1) \text{(13)} \qquad \\ \sigma_{31} & = \mu\alpha(\psi_{,1} - x_1) \text{(14)} \qquad \end{align}$$

Equilibrium

\sigma_{ji,j} = 0 ~ \text{no body forces.} $$ Therefore,
 * $$\begin{align}

\sigma_{11,1} + \sigma_{21,2} + \sigma_{31,3} = 0 & \Rightarrow 0 = 0 \\ \sigma_{12,1} + \sigma_{22,2} + \sigma_{32,3} = 0 & \Rightarrow 0 = 0 \\ \sigma_{13,1} + \sigma_{23,2} + \sigma_{33,3} = 0 & \Rightarrow \mu\alpha(\psi_{,11}+\psi_{,22}) = \mu\alpha\nabla^2{\psi} = 0 \text{(15)} \qquad \end{align}$$

Internal Tractions

 * Normal to cross sections is $$\widehat{\mathbf{n}}{} = \widehat{\mathbf{e}}{3}\,$$.
 * Normal traction $$t^n = \mathbf{t}\bullet\widehat{\mathbf{n}}{} = 0\,$$.
 * Projected shear traction is  $$t^s = \sqrt{\sigma_{13}^2 + \sigma_{23}^2}\,$$.
 * Traction vector at a point in the cross section is tangent to the cross section.

Boundary Conditions on Lateral Surfaces
We parameterize the boundary curve $$\partial S$$ using
 * Lateral surface traction-free.
 * Unit normal to lateral surface appears as an in-plane unit  normal to the boundary $$\partial S$$.

\mathbf{x} = \tilde{\mathbf{x}}(s) ~, 0 \le s \le l; \tilde{\mathbf{x}}(0) = \tilde{\mathbf{x}}(l) $$ The tangent vector to $$s$$ is

\widehat{\boldsymbol{\nu}} = \frac{d\mathbf{x}}{ds} ~\text{and} \widehat{\mathbf{n}}{} = \widehat{\boldsymbol{\nu}}\times\widehat{\mathbf{e}}_{3} \Rightarrow \widehat{\mathbf{n}}{} = \frac{dx_2}{ds} \widehat{\mathbf{e}}{1} - \frac{dx_1}{ds} \widehat{\mathbf{e}}_{2} $$ The tractions $$t_1\,$$ and $$t_2\,$$ on the lateral surface are identically zero. However, to satisfy the BC $$t_3 = 0\,$$, we need

t_3 = n_1 \sigma_{13} + n_2 \sigma_{23} = 0 \Rightarrow \left(\psi_{,1} - x_2\right) n_1 + \left(\psi_{,2} + x_1\right) n_2= 0 $$ or,
 * $$ \text{(16)} \qquad

\left(\psi_{,1} - x_2\right) \frac{dx_2}{ds} + \left(\psi_{,2} + x_1\right) \frac{dx_1}{ds}= 0 $$

Boundary Conditions on End Surfaces
The traction distribution is statically equivalent to the torque $$\mathbf{T}\,$$. At $$x_3 = L\,$$,

t_1 = \sigma_{13}~; t_2 = \sigma_{23}~; t_3 = \sigma_{33} = 0 \, $$ Therefore,

F_1 = \int_S \sigma_{13}~dS = \mu\alpha\int_S(\psi_{,1}-x_2)~dS $$ From equilibrium,
 * $$\begin{align}

\nabla^2{\psi} = 0 \Rightarrow \psi_{,1}-x_2 & = (\psi_{,1}-x_2) + x_1(\psi_{,11} + \psi_{,22}) \\ & = \psi_{,1} + x_1\psi_{,11} - x_2 + x_1\psi_{,22} \\ & = (x_1\psi_{,1} - x_1x_2)_{,1} + (x_1\psi_{,2} + x_1x_1)_{,2} \\ & = \left[x_1(\psi_{,1} - x_2)\right]_{,1} + \left[x_1(\psi_{,2} + x_1)\right]_{,2} \end{align}$$ Hence,
 * $$\text{(17)} \qquad

F_1 = \mu\alpha\int_S\left[x_1(\psi_{,1} - x_2)\right]_{,1} + \left[x_1(\psi_{,2} + x_1)\right]_{,2} dS $$

The Green-Riemann Theorem
If $$P = f(x_1,x_2)\,$$ and $$Q = q(x_1,x_2)\,$$ then

\int_S (Q_{,1} - P_{,2}) dS = \oint_{\partial S} (P dx_1 + Q dx_2) $$ with the integration direction such that $$S$$ is to the left.

Applying the Green-Riemann theorem to equation (17), and using equation (16)
 * $$\text{(18)} \qquad

F_1 = \mu\alpha\oint_{\partial S} -x_1(\psi_{,2} + x_1)dx_1 + x_1(\psi_{,1} - x_2)dx_2 = 0 $$ Similarly, we can show that $$F_2 = 0\,$$. $$F_3 = 0\,$$ since $$t_3 = 0\,$$.

The moments about the $$x_1\,$$ and $$x_2\,$$ axes are also zero.

The moment about the $$x_3\,$$ axis is

M_3 = \int_S (x_1\sigma_{23} - x_2\sigma_{13}) dS = \mu\alpha\int_S(x_1\psi_{,2} + x_1^2 - x_2\psi_1 + x_2^2) dS = \mu\alpha\tilde{J} $$ where $$J\,$$ is the torsion constant. Since $$M_3 = T\,$$, we have

\alpha = \frac{T}{\mu\tilde{J}} $$ If $$\psi = 0\,$$, then $$\tilde{J} = J\,$$, the polar moment of inertia.

Summary of the solution approach

 * Find a warping function $$\psi(x_1,x_2)\,$$ that is harmonic. and satisfies the traction BCs.
 * Compatibility is not an issue since we start with displacements.
 * The problem is independent of applied torque and the material properties of the cylinder.
 * So it is just a geometrical problem. Once $$\psi$$ is known, we can calculate
 * The displacement field.
 * The stress field.
 * The twist per unit length.


text:
Elasticity/Warping functions

Warping Function and Torsion of Non-Circular Cylinders
Warping functions are quite useful in the solution of problems involving the torsion of cylinders with non-circular cross sections.

For such problems, the displacements are given by

u_1 = -\alpha x_2 x_3 ~; u_2 = \alpha x_1 x_3 ~; u_3 = \alpha\psi(x_1,x_2) $$ where $$\alpha\,$$ is the twist per unit length, and $$\psi\,$$ is the warping function.

The stresses are given by

\sigma_{13} = \mu\alpha(\psi_{,1} - x_2) ~; \sigma_{23} = \mu\alpha(\psi_{,2} + x_1) $$ where $$\mu\,$$ is the shear modulus.

The projected shear traction is

\tau = \sqrt{(\sigma_{13}^2 + \sigma_{23}^2)} $$

Equilibrium is satisfied if

\nabla^2{\psi} = 0 ~ \forall (x_1, x_2) \in \text{S} $$

Traction-free lateral BCs are satisfied if

(\psi_{,1} - x_2) \frac{dx_2}{ds} - (\psi_{,2} + x_1) \frac{dx_1}{ds} = 0 ~ \forall (x_1, x_2) \in \partial\text{S} $$ or,

(\psi_{,1} - x_2) \hat{n}_{1} + (\psi_{,2} + x_1) \hat{n}_{2} = 0 ~ \forall (x_1, x_2) \in \partial\text{S} $$

The twist per unit length is given by

\alpha = \frac{T}{\mu \tilde{J}} $$ where the torsion constant

\tilde{J} = \int_S (x_1^2 + x_2^2 + x_1\psi_{,2} - x_2\psi_{,1}) dA $$

title: Elasticity/Warping functions


text:
Elasticity/Warping of circular cylinder

Example 1: Circular Cylinder
Choose warping function

\psi(x_1,x_2) = 0 \, $$

Equilibrium ($$\nabla^2{\psi} = 0$$) is trivially satisfied.

The traction free BC is

(0 - x_2) \frac{dx_2}{ds} - (0 + x_1) \frac{dx_1}{ds} = 0 ~ \forall (x_1, x_2) \in \partial\text{S} $$ Integrating,

x_2^2 + x_1^2 = c^2 ~ \forall (x_1, x_2) \in \partial\text{S} $$ where $$c$$ is a constant.

Hence, a circle satisfies traction-free BCs.

 * There is no warping of cross sections for circular cylinders

The torsion constant is

\tilde{J} = \int_S (x_1^2 + x_2^2) dA = \int_S r^2 dA = J $$

The twist per unit length is

\alpha = \frac{T}{\mu J} $$

The non-zero stresses are

\sigma_{13} = -\mu\alpha x_2 ~; \sigma_{23} = \mu\alpha x_1 $$

The projected shear traction is

\tau = \mu\alpha\sqrt{(x_1^2 + x_2^2)} = \mu\alpha r $$

Compare results from Mechanics of Materials solution

\phi = \frac{TL}{GJ} \Rightarrow \alpha = \frac{T}{GJ} $$ and

\tau = \frac{Tr}{J} \Rightarrow \tau = G\alpha r $$

title: Elasticity/Warping of circular cylinder


text:
Elasticity/Warping of elliptical cylinder

Example 2 : Elliptical cylinder
Choose warping function

\psi(x_1,x_2) = k x_1 x_2 \, $$ where $$k\,$$ is a constant.

Equilibrium ($$\nabla^2{\psi} = 0$$) is satisfied.

The traction free BC is

(kx_2 - x_2) \frac{dx_2}{ds} - (kx_1 + x_1) \frac{dx_1}{ds} = 0 ~ \forall (x_1, x_2) \in \partial\text{S} $$ Integrating,

x_1^2 + \frac{1-k}{1+k}x_2^2 = a^2 ~ \forall (x_1, x_2) \in \partial\text{S} $$ where $$a\,$$ is a constant.

This is the equation for an ellipse with major and minor axes $$a\,$$ and $$b\,$$, where

b^2 = \left(\frac{1+k}{1-k}\right) a^2 $$

The warping function is

\psi = -\left(\frac{a^2-b^2}{a^2 + b^2}\right) x_1 x_2 $$

The torsion constant is

\tilde{J} = \frac{2b^2}{a^2+b^2} I_2 + \frac{2a^2}{a^2+b^2} I_1 = \frac{\pi a^3 b^3}{a^2 + b^2} $$ where

I_1 = \int_S x_1^2 dA = \frac{\pi a b^3}{4} ; I_2 = \int_S x_2^2 dA = \frac{\pi a^3 b}{4} $$

If you compare $$\tilde{J}$$ and $$J\,$$ for the ellipse, you will find that $$\tilde{J} < J$$. This implies that the torsional rigidity is less than that predicted with the assumption that plane sections remain plane.

The twist per unit length is

\alpha = \frac{(a^2+b^2)T}{\mu\pi a^3 b^3} $$

The non-zero stresses are

\sigma_{13} = -\frac{2\mu\alpha a^2 x_2}{a^2 + b^2} ; \sigma_{23} = -\frac{2\mu\alpha b^2 x_1}{a^2 + b^2} $$

The projected shear traction is

\tau = \frac{2\mu\alpha}{a^2+b^2}\sqrt{b^4 x_1^2 + a^4 x_2^2} \Rightarrow \tau_{\text{max}} = \frac{2\mu\alpha a^2 b}{a^2+b^2} (b < a) $$

For any torsion problem where $$\partial S\,$$ is convex, the maximum projected shear traction occurs at the point on $$\partial S\,$$ that is nearest the centroid of $$S\,$$.

The displacement $$u_3\,$$ is

u_3 = -\frac{(a^2-b^2)Tx_1x_2}{\mu\pi a^3b^3} $$


text: File:Subselection tool.jpg

title: File:Subselection tool.jpg


text:
Elasticity/Warping of rectangular cylinder

Example 3: Rectangular Cylinder
In this case, the form of $$\psi\,$$ is not obvious and has to be derived from the traction-free BCs

(\psi_{,1} - x_2) \hat{n}_{1} + (\psi_{,2} + x_1) \hat{n}_{2} = 0 ~ \forall (x_1, x_2) \in \partial\text{S} $$

Suppose that $$2a\,$$ and $$2b\,$$ are the two sides of the rectangle, and $$a > b\,$$. Also $$a\,$$ is the side parallel to $$x_1\,$$ and $$b\,$$ is the side parallel to $$x_2\,$$. Then, the traction-free BCs are

\psi_{,1} = x_2 \text{on} x_1 = \pm a ,\text{and} \psi_{,2} = -x_1 \text{on} x_2 = \pm b \,$$

A suitable $$\psi\,$$ must satisfy these BCs and $$\nabla^2{\psi} = 0\,$$.

We can simplify the problem by a change of variable

\bar{\psi} = x_1~x_2 - \psi $$

Then the equilibrium condition becomes

\nabla^2{\bar{\psi}} = 0 $$

The traction-free BCs become

\bar{\psi}_{,1} = 0 \text{on} x_1 = \pm a ,\text{and} \bar{\psi}_{,2} = 2x_1 \text{on} x_2 = \pm b $$

Let us assume that

\bar{\psi}(x_1,x_2) = f(x_1) g(x_2) $$

Then,

\nabla^2{\bar{\psi}} = \bar{\psi}_{,11} + \bar{\psi}_{,22} = f^{}(x_1) g(x_2) + g^{}(x_2) f(x_1) = 0 $$ or,

\frac{f^{}(x_1)}{f(x_1)} = - \frac{g^{}(x_2)}{g(x_2)} = \eta $$

Case 1: η > 0 or η = 0
In both these cases, we get trivial values of $$C_1 = C_2 = 0\,$$.

Case 2: η < 0
Let

\eta = -k^2 ; k > 0 $$

Then,
 * $$\begin{align}

f^{''}(x_1) + k^2 f(x_1) = 0 \Rightarrow & f(x_1) = C_1 \cos(kx_1) + C_2 \sin(kx_1) \\ g^{''}(x_2) - k^2 g(x_2) = 0 \Rightarrow & g(x_2) = C_3 \cosh(kx_2) + C_4 \sinh(kx_2) \end{align}$$

Therefore,

\bar{\psi}(x_1,x_2) = \left[ C_1 \cos(kx_1) + C_2 \sin(kx_1) \right] \left[ C_3 \cosh(kx_2) + C_4 \sinh(kx_2) \right] $$

Apply the BCs at $$x_2 = \pm b\,$$ ($$\bar{\psi}_{,2} = 2x_1$$), to get
 * $$\begin{align}

\left[ C_1 \cos(kx_1) + C_2 \sin(kx_1) \right] \left[ C_3 \sinh(kb) + C_4 \cosh(kb) \right] & = 2x_1\\ \left[ C_1 \cos(kx_1) + C_2 \sin(kx_1) \right] \left[-C_3 \sinh(kb) + C_4 \cosh(kb) \right] & = 2x_1 \end{align}$$ or,

F(x_1) G^{'}(b) = 2 x_1 ; F(x_1) G^{'}(-b) = 2 x_1 \, $$ The RHS of both equations are odd. Therefore, $$F(x_1)$$ is odd. Since, $$\cos(k x_1)\,$$ is an even function, we must have $$C_1 = 0\,$$.

Also,

F(x_1) \left[ G^{'}(b) - G^{'}(-b)\right] = 0 $$ Hence, $$G'(b)\,$$ is even. Since $$\sinh(kb)\,$$ is an odd function, we must have $$C_3 = 0\,$$.

Therefore,

\bar{\psi}(x_1,x_2) = C_2 C_4 \sin(kx_1) \sinh(kx_2) = A \sin(kx_1) \sinh(kx_2) $$

Apply BCs at $$x_1 = \pm a\,$$ ($$\bar{\psi}_{,1} = 0$$), to get

A k \cos(ka) \sinh(kx_2) = 0 \, $$ The only nontrivial solution is obtained when $$\cos(ka) = 0$$, which means that

k_n = \frac{(2n+1)\pi}{2a}, n = 0,1,2,... $$

The BCs at $$x_1 = \pm a\,$$ are satisfied by every terms of the series

\bar{\psi}(x_1,x_2) = \sum_{n=0}^{\infty} A_n \sin(k_n x_1) \sinh(k_n x_2) $$

Applying the BCs at $$x_1 = \pm b\,$$ again, we get

\sum_{n=0}^{\infty} A_n k_n \sin(k_n x_1) \cosh(k_n b) = 2 x_1 \Rightarrow \sum_{n=0}^{\infty} B_n \sin(k_n x_1) = 2 x_1 $$

Using the orthogonality of terms of the sine series,

\int_{-a}^a \sin(k_n x_1) \sin(k_m x_1) dx_1 = \begin{cases} 0 & {\rm if}~ m \ne n \\ a & {\rm if}~ m = n \end{cases} $$ we have

\int_{-a}^a \left[\sum_{n=0}^{\infty} B_n \sin(k_n x_1)\right] \sin(k_m x_1) dx_1 = \int_{-a}^a \left[2 x_1\right] \sin(k_m x_1) dx_1 $$ or,

B_m a = \frac{4}{a k_m^2} \sin(k_m a) $$ Now,

\sin(k_m a) = \sin\left(\frac{(2m+1)\pi}{2}\right) = (-1)^m $$ Therefore,

A_m = \frac{B_m}{k_m\cosh(k_m b)} = \frac{(-1)^m 32a^2}{(2m+1)^3\pi^3\cosh(k_m b)} $$

The warping function is

\psi = x_1 x_2 - \frac{32a^2}{\pi^3} \sum_{n=0}^{\infty} \frac{(-1)^n \sin(k_n x_1) \sinh(k_n x_2)}{(2n+1)^3\cosh(k_n b)} $$ The torsion constant and the stresses can be calculated from $$\psi$$.