TEP Aula 01 (12032013) - Prof. Nilo C. Consoli

7/28/2019 TEP Aula 01 (12032013) - Prof. Nilo C. Consoli

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l l l _ f _ I ! g L n ~ . . r i n s .. < i P P J i c < t \ Q I . J S _ . l i ! . < ; J ; s p o n s c . Q . \ i . b y ~ _ ( \ } " __ ~ ! . ! _ I < : ! i c d - i : f ~ ~ - C T O ~ . o p j _ ~ - J ~ ~ ~ _ I

\lithoul-considcring a tomic-and molecular s t r u c t u r ~ - The ~ u b j ~ L . 9 L ~ 1 ~ \ b i n g _m;J.tc;rial .bc.h.aviQr. aLJ.hun

Sludvf the response of a substance or body under external excitationc q ~ j t u t ~ s th9 W(:tjor ~ n d e a v o r in engineering an d sciences. Tiielinpj)rt:i_ryj_i n g ~ ~ t s i p y o l v ~ d in ~ u c h a st.udy_ar_..(a) crnal cxcitation...(b)?hs internal" n titutJon of the medrum an c the -------=-

... ll .

F

Figure I-I P l ~ c c of consli:,,J;:c bws in wntinuum rncch;mics.

tion hehavior of the body.As another e x ~ m p l c , consider a pool of water ill1d a ~ : > o l of icc. Thechemical a : - ~ d at01i1ic structures of these two bodies are the s:1me. we Qnsee. water and ice will respond c hclu1ve '1 two difl'crcnt w;J,y;; lilldcr the sameexternal loads. Although their chemka! and atomjc stflJ';tUies are the sa.mc,the internal c-.nstitution of matter for water and ice is di..ITercnL This e o n s t i t u ~tion of matter governs the behavior of the body. Th e rclatioru.rup betweencause and effect can be cniled the consriwtice /all' of the material for a gin:nphenomenon.

THEORY OF CONSTITUTIVE EQUATIONSreduction


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NOTACAO:

1 FYa!Y= un-dAdA => 0

lim Fza ~ - =dAd.'l => 0 T ~ n s o e s Cisalhantes Cixy, Ciu. ...

CONVENCOES (Mec. Solos)Tensoes Normais Compressivas sao positivas (direcionadas para dentroda superficie) Tensoes Cisalhantes sao positivas dirigidas na direvao dos eixoscartesianos enquanto agindo em um plano no qual a tensao normal esta na mesmadirevao dos eixos cartesianos. Se a tensao normal for contniria ao eixo cartesiano, as

tensoes cisalhantes positivas serao no sentido contnirio aos eixos cartesianos.

z


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CONCEITO DE TENSAO

n

Tensao (u) . dFcr = hm =- dAUA=>O

Dcslocrunentos (3 grandeMIS a serQID de1lilldas)D e f u r m a ~ O e s (6 gnmdcMIS a ser 1 dcllnldas)Tenst>es (6 grandezm; a serem definictas)TOTAL : (1.5 granill;zas a serem defmidas)

--------=---=-)

~lD ' { ' (

- - - -------

) \T83


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\..- Ib ~ t..QOI Ll r ~ P , \0

The equilibrium of au dement requires also that the resultant momentvanish. I f there do not exist extemalmoments proportional to a volume, theconsideration of moments will lead to the important conclusion that thestress tensor is symmetric,

II .__ _ _II XzI -4--,

, e . . . . - - - ~ \ = - - - 1 t'31 1I 1 chu1 4 - - - 1 ~ 1,_ _ . , . Tu + o;;dx1: X1 tt'32J- ...1

~ Components of tractions that contribute moment about Ox,-altil.moment of all the forces about the x 3-axis, we see that those components offorces parallel to Ox 3 or lying in planes containing Ox, do not contributeany moment. The components that do contribute a moment about tbe x,axisare shown in Fig. 3.9. Therefore, properly taking care of the moment arm,we have

( + 01:11 d ) d d dXz 1 d d dx2- T 11 C1X; x 1 x 2 x3T-rt 11 Xz x 3 -y+ (r: 11 + dx,) dx 2 dx 3 dx 1 - (r 21 + dx 2) dx, dx, dx 21_ ( + 1:U d ) d d dx 1 d d aX 1-r ru a i; Xz. x1 x3T- T 22 x, x3 T+ ( +dr:n.d ) d d dx 1 d d dx 1fn ax, XJ .-c,, Xz T- t'32 x , Xz T

( + Of,, l ) d d dx z '- d d dx 2. - t' 31 "JX'; Gx3 x 1 x 2 T -r t 31 X1 Xz T. - x, dx, dxl dx, d? + xl dx , dxz dx, = 0.

On dividing through by dx 1 dx 2 dx 3 and passing to the limit dx 1 - 0,dx 2 - 0, dx 3 - 0, we obtain

Te /JS)SP( AS OvTfl.IJ$

rG"!o 0 (;?o-:]r ~ : ~ G,z L

0


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Fig. 3.8 Compommts or tractions in XJdirection.force and one component of body force. The sum is(r 11 + dx 1) dx 2 dx 3 - T 11 dx 2 dx 3 + (-r 21 + dx:) dx 3 dx 1

- T21 dx3 dx 1+ (r31 + dx 3) dx 1 dx:- 't'31 dx 1 dx2 + X 1 dx 1 dxz dx 3 = 0.

Dividing by dx 1 dx2 dx3, we obtain

, ..JDl R qvcs.

(]. . . _ - . ) ?c/ OJ?J

d ~ j- - - - - - - - - - - - - - - - - - - . . - - ~ - - ... --

+.. .. . -- . .. .


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1.3 EguiHbrlWJl ~ Q Y l a nl . 3 . l GARJES,Ati CO.ORP lNAJESBy Qonsidering the equilibrium of th e elementS:hQW!l i l l Fig.l . l in the Cartesian coordinate system,the fQllowing equilibrium equations are obtained:

. . (1 . 28a)

. . . (1 . 28b)

where X, Y, Z are the body forces, perunit volume, in th e x, y andz directions.With an ordinuy gravity field and th e z direction vertically downwards, X and Y are zero and

z. is th e unit weight, y, of the material.

1.3.2 CYLINDRICAL COORDINATES

0

Pion of


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4.3 OESLOCAMENTOS E OEFORMA


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(7) [Fig. 5-4, j. This device permit$testing of a cylindrical specimen of a material. A confining stress (pressure)equal to o3 ( = a2 ) is applied, usually by using a _luid in the chamber. In thecase of geologic media, this c o n ~ n i n g pressure sirritJates initial or in situstresses that exist at the site before a oad increment is.applied. The axial (ol)or deviator stress; (a 1 - a3), is then applied, which cause> shearing of .the

S i l l l l p ,M ~ ~ l J J ~ e n t s are obtained in terms of the two'stresses, a 1, a2 = o3 , axialf o r m a t i o n (strah), l a t ~ r a l or rad,ial _deforination (strain), - ____ These measurements allow plotting of resultsiii-

various forms; typical schematic plots in terms of stress difference (a 1 - o3 )versus axial strain (c1) and radial strain (( 3) versus (( 1) are shown in .Fig,\ ,

'1Confiningfluid

Pore_ pressure (b )

. (a)i Figure ~ Cylindrical triaxial t.esl ,

IJ ~ l . i n n rIIIa II

III

Not li car (non'ine r)

Symbolic stress-strrunrelation. _

i

....__ E.----'- ~ G ; J _ l,jjj _ _ __ .JIC,l


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1000- 900_~ 800...._,Q) 700uQQ) 600-1

~~- 500Q


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20

1000- ~ ~ ; j ' )_g ~ ~800.!oi:..._,13)g~ 600!B'-+-

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TEP Aula 01 (12032013) - Prof. Nilo C. Consoli