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1MathematicalPreliminaries

Similartootherfieldtheoriessuchasfluidmechanics,heatconduction,andelectromagnetics,

thestudyandapplicationofelasticitytheoryrequiresknowledgeofseveralareasofapplied

mathematics.Thetheoryisformulatedintermsofavarietyofvariablesincludingscalar,

vector,andtensorfields,andthiscallsfortheuseoftensornotationalongwithtensoralgebra

andcalculus.Throughtheuseofparticularprinciplesfromcontinuummechanics,thetheoryis

developedasasystemofpartialdifferentialfieldequationsthataretobesolvedinaregionof

spacecoincidingwiththebodyunderstudy.Solutiontechniquesusedonthesefieldequations

commonlyemployFouriermethods,variationaltechniques,integraltransforms,complex

variables,potentialtheory,finitedifferences,andfiniteandboundaryelements.Therefore,to

developproperformulationmethodsandsolutiontechniquesforelasticityproblems,itis

necessarytohaveanappropriatemathematicalbackground.Thepurposeofthisinitialchapter

istoprovideabackgroundprimarilyfortheformulationpartofourstudy.Additionalreviewof

othermathematicaltopicsrelatedtoproblemsolutiontechniqueisprovidedinlaterchapters

wheretheyaretobeapplied.

1.1Scalar,Vector,Matrix,andTensorDefinitions

Elasticitytheoryisformulatedintermsofmanydifferenttypesofvariablesthatareeither

specifiedorsoughtatspatialpointsinthebodyunderstudy.Someofthesevariablesarescalar

quantities,representingasinglemagnitudeateachpointinspace.Commonexamplesinclude

thematerialdensityrandmaterialmodulisuchasYoung’smodulusE,Poisson’sration,or

theshearmodulusm.Othervariablesofinterestarevectorquantitiesthatareexpressiblein

termsofcomponentsinatwo-orthree-dimensionalcoordinatesystem.Examplesofvector

variablesarethedisplacementandrotationofmaterialpointsintheelasticcontinuum.

Formulationswithinthetheoryalsorequiretheneedformatrixvariables,whichcommonly

requiremorethanthreecomponentstoquantify.Examplesofsuchvariablesincludestressand

strain.Asshowninsubsequentchapters,athree-dimensionalformulationrequiresnine

components(onlysixareindependent)toquantifythestressorstrainatapoint.Forthis

case,thevariableisnormallyexpressedinamatrixformatwiththreerowsandthreecolumns.

Tosummarizethisdiscussion,inathree-dimensionalCartesiancoordinatesystem,scalar,

vector,andmatrixvariablescanthusbewrittenasfollows:

3massdensityscalar¼r

displacementvector¼u¼ue

1þve

2þwe

3

stressmatrix¼[s]¼s

xt

xyt

xz

t

yxs

yt

yz

t

zxt

zys

z2

6

43

7

5

wheree

1,e

2,e

3aretheusualunitbasisvectorsinthecoordinatedirections.Thus,scalars,

vectors,andmatricesarespecifiedbyone,three,andninecomponents,respectively.

Theformulationofelasticityproblemsnotonlyinvolvesthesetypesofvariables,butalso

incorporatesadditionalquantitiesthatrequireevenmorecomponentstocharacterize.Because

ofthis,mostfieldtheoriessuchaselasticitymakeuseofatensorformalismusingindexnotation.

Thisenablesefficientrepresentationofallvariablesandgoverningequationsusinga

singlestandardizedscheme.Thetensorconceptisdefinedmorepreciselyinalatersection,

butfornowwecansimplysaythatscalars,vectors,matrices,andotherhigher-ordervariables

canallberepresentedbytensorsofvariousorders.Wenowproceedtoadiscussiononthe

notationalrulesoforderforthetensorformalism.Additionalinformationontensorsandindex

notationcanbefoundinmanytextssuchasGoodbody(1982)orChandrasekharaiahand

Debnath(1994).

1.2IndexNotation

Indexnotationisashorthandschemewherebyawholesetofnumbers(elementsorcompon-

ents)isrepresentedbyasinglesymbolwithsubscripts.Forexample,thethreenumbers

a

1,a

2,a

3aredenotedbythesymbola

i,whereindexiwillnormallyhavetherange1,2,3.

Inasimilarfashion,a

ijrepresentstheninenumbersa

11,a

12,a

13,a

21,a

22,a

23,a

31,a

32,a

33.

Althoughtheserepresentationscanbewritteninanymanner,itiscommontouseascheme

relatedtovectorandmatrixformatssuchthat

a

i¼a

1

a

2

a

32

43

5,a

ij¼a

11a

12a

13

a

21a

22a

23

a

31a

32a

332

43

5(1:2:1)

Inthematrixformat,a

1jrepresentsthefirstrow,whilea

i1indicatesthefirstcolumn.Other

columnsandrowsareindicatedinsimilarfashion,andthusthefirstindexrepresentstherow,

whilethesecondindexdenotesthecolumn.

Ingeneralasymbola

ij...kwithNdistinctindicesrepresents3Ndistinctnumbers.It

shouldbeapparentthata

ianda

jrepresentthesamethreenumbers,andlikewisea

ijand

a

mnsignifythesamematrix.Addition,subtraction,multiplication,andequalityofindex

symbolsaredefinedinthenormalfashion.Forexample,additionandsubtractionare

givenby

a

iÆb

i¼a

1Æb

1

a

2Æb

2

a

3Æb

32

43

5,a

ijÆb

ij¼a

11Æb

11a

12Æb

12a

13Æb

13

a

21Æb

21a

22Æb

22a

23Æb

23

a

31Æb

31a

32Æb

32a

33Æb

332

43

5(1:2:2)

andscalarmultiplicationisspecifiedas

4FOUNDATIONSANDELEMENTARYAPPLICATIONS

合集下载

SBS朗文国际英语教程第1册-教师用书

SBS朗文国际英语教程第1册-教师用书

FUNCTIONS

ASKINGFORANDREPORTINGINFORMATION

What’s your name?

My name is Maria.

What’s your first name?

Sarah.

What’s your last name?

Kelly.

How do you spell that?

K-E-L-L-Y.

What’s your address?

My address is 235 Main Street.

What’s your phone number?

My phone number is 741-8906.

Where are you from?

I’m from Mexico City.

I’m American.

My license number is 921DCG.

My apartment number is 4-B.

Mysocial security number is

044-35-9862.

Mye-mail address is

TeacherJoe@

.

CHAPTER 1CHAPTER 1 OVERVIEW:Text Pages 1–6

2MEETINGPEOPLE

Hello.

Hi.

My name is_______.

I’m_______.

Nice to meet you.

Nice to meet you,

too.TOBE

I am from Mexico City.

(I am)

What’s your name?

(What is)

My name is Maria.

Where are you from?am

is

areGRAMMARCHAPTER 1

3NEW VOCABULARY

People

actor

actress

American

athlete

Mr.

Mrs.

Ms.

president

prime minister

Places

Brooklyn

Florida

Mexico City

渔夫和他灵魂1至3章读后感英文版

渔夫和他灵魂1至3章读后感英文版

渔夫和他灵魂1至3章读后感英文版

全文共6篇示例，供读者参考

篇1

Title: The Fisherman and His Soul - My Thoughts

Introduction:

Hi everyone! Today, I want to share my thoughts about a

really cool book called "The Fisherman and His Soul." I just

finished reading the first three chapters, and it was so exciting!

Let me tell you all about it.

Chapter 1 - The Fisherman's Discovery:

In the first chapter, we meet the fisherman. He's a

kind-hearted person who loves fishing. One day, he catches a

beautiful fish, but something magical happens! The fish can talk

and asks the fisherman to let it go. I was surprised because I've

never heard of a talking fish before! The fisherman decides to

release the fish, and they become friends. It made me happy to

see the fisherman's kindness.

Chapter 2 - The Soul's Adventure: The second chapter was even more fascinating! The

美国海岸工程手册Part_I-Chap_1

美国海岸工程手册Part_I-Chap_1

IntroductionI-1-iChapter 1EM 1110-2-1100

INTRODUCTION(Part I)

1 August 2008 (Change 2)

Table of Contents

Page

I-1-1. Purpose and Scope.......................................................I-1-1

I-1-2. Applicability.............................................................I-1-1

I-1-3. Definitions...............................................................I-1-1

a. Coastal.................................................................I-1-1

b. Coastal engineering.......................................................I-1-1

c. Coastal science..........................................................I-1-2

I-1-4. Bibliography.............................................................I-1-2

I-1-5. References...............................................................I-1-2

I-1-6. Acknowledgments........................................................I-1-4EM 1110-2-1100 (Part I)

语言学第一章

语言学第一章

语言学第一章

Chapter 1 Invitation to linguistics

1.1 Why study language?

1. Language is very essential to human beings.

2. In language there are many things we should know.

3. For further understanding, we need to study language

scientifically.

1.2 What is language?

Language is a means of verbal communication. It is a system

of arbitrary vocal symbols used for human communication.

1.3 Design features of language

The features that define our human languages can be called

design features which can distinguish human language from any

animal system of communication.

1.3.1 Arbitrariness

Arbitrariness refers to the fact that the forms of linguistic

signs bear no natural relationship to their meanings.

1.3.2 Duality

Duality refers to the property of having two levels of

structures, such that units of the primary level are composed of