ME333B Elasticity and Inelasticity – Wei Cai – Stanford University – Win 2017

Handout 1. Introduction to Elasticity Theory

February 12, 2017

1 Scope of treatment

Elasticity theory determines stress and displacement in a body as a result of applied (me-chanical or thermal) load. — J. R. Barber, “Elasticity”, 3rd ed. (2009)

A body under elastic deformation reverts to its original state when the applied load is re-moved.

Elasticity theory is also important for the study of inelastic deformation, such as fracture andplasticity, by studying the microscopic agents of inelasticity, such as a crack or a dislocation.The application of elasticity theory to microscopic defects in solids is called micromechanics.

In the elasticity section of this course, we will focus on linear, infinitesimal elasticity. Thismeans that the stress and displacements increase linearly with applied loads. The deforma-tion is assumed to be so small that we assume linear superposition can be used to constructa new solution based on two existing solutions. For simplicity, we shall also assume that thebody is in static equilibrium, i.e. no dynamical e↵ects will be considered.

Elasticity theory goes a step beyond the undergraduate course on “Mechanics of Materials”,in which certain assumptions (e.g. regarding the deformation of a beam) are made to keepthe analysis simple. In elasticity theory, a more rigorous mathematical formulation is es-tablished to describe the deformation of a solid (based on partial di↵erential equations), so

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that the simplifying assumptions in “Mechanics of Materials” are no longer needed. Fur-thermore, using elasticity theory, we can see whether or when these simplifying assumptionsare justified.

We shall see that various approximations are still invoked in elasticity theory, especially whentrying to obtain an analytic solution for a given problem. However, these approximationstend to be made towards the end of the analysis, when trying to obtain the final solution. Theapproximations also tend to be mathematical (rather than physical) in nature. In addition,elasticity theory also provides ways to quantify the error invoked in these approximations,and to develop corrective solutions to further reduce the error, if desired.

A number of mathematical tools will be used in this course, including tensor algebra, par-tial di↵erential equations, and Fourier transform. Some of these tools may be beyond thecurriculum of an undergraduate engineering program. Introductions to these tools will begiven when appropriate.

Outline of the elasticity section:

1. Fundamental variables and equations of elasticity. Variables: stress, strain, displace-ment, traction force, elastic constants. Equations: equilibrium, compatibility, Gener-alized Hooke’s law.

2. Methods of solution and applications. Methods: Airy stress function, Green function,Fourier transform. Applications: thick beam, contact, crack, dislocation.

While we will try to develop analytic solutions, Matlab will be used to visualize the solution,as well as to compare the analytic solutions with numerical solutions from finite elementmethod (FEM). We will also use Matlab to assist with the symbolic manipulations requiredin developing analytic solutions.

Relationship with other courses

• Undergraduate “Mechanics of Materials” course is a pre-requisite. Basic knowledge ofstress and strain is required before taking this course.

• Students already exposed to “Continuum Mechanics” will benefit from the similar no-tations and assumptions. But “Continuum Mechanics” is not a pre-requisite for thiscourse. In this course, we will make simplifying assumptions of linear and infinitesi-mal elasticity so that the full machinary of continuum mechanics (applicable to largedeformations) is not needed. Compared with continuum mechanics, the focus of thiscourse is more on the physical interpretation of the solutions than on the mathematicalformulation.

• Many elasticity (and inelasticity) problems today are solved by numerical methods suchas the “Finite Element Method” (FEM). Elasticity theory provides the fundamentalequations to be solved by the numerical methods. The analytic solutions obtained from

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elasticity theory also reveal important physics that are di�cult to see by numericalmethods. The analytic solutions (such as the stress singularity at the crack tip) alsomotivate and assist the development of more e�cient numerical methods.

Textbook

• J. R. Barber, “Elasticity”, 3rd ed. 2009. PDF available through Stanford library (seeCanvas)

First Reading Assignment

• Hibbeler “Mechanics of Materials”, § 3.2, available on Canvas/Materials/Extra Read-ing.

• (optional) Mechanical Properties Table of Typical Engineering Materials from Hi-bbeler, available on Canvas.

• (optional) Experimental stress-strain curves of pure copper from Kocks and Mecking,available on Canvas.

2 Examples

The following diagrams show typical stress-strain curves of ductile and brittle materials whenloaded in uniaxial tension.

Please see Hibbeler (reading assignment above) for definitions of:

• Young’s modulus E

• Yield stress �Y

• Ultimate strength �u

• Proportional limit �p

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We now consider in more detail how much stress is generated if a rectangular block of materialis loaded in uniaxial tension along the z-axis.

The left side of the above diagram shows a rectangular block made of a homogeneous materialwith Young’s modulus E. Let the elongation of the block be �L in response to the appliedtensile force F . The normal stress strain along the z-axis are:

• (axial) stress: �zz = F/A (where A = a · b)

• (axial) strain: "zz = �L/L

Below the proprotional limit (�zz < �p

), the stress and strain are linear to each other,�zz = E "zz. From the above relations, we can obtain the familiar result in Mechanics ofMaterials:

�L =F L

E A(1)

Let us now consider the case shown on the right side of the above diagram, where the platecontains a circular hole of radius r. The area of the cross section containing the center ofthe circular hole is,

A0 = (a� 2r) · b (2)

The average normal stress on this cross section is

(�zz)avg = F/A0 (3)

However, from Mechanics of Materials, we expect that the stress distribution on this crosssection is not uniform, and that the maximum stress will be greater than the average stress.This phenomenon is described by the stress concentration factor K, so that

(�zz)max

= K · (�zz)avg = K · F/A0 (4)

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In Mechanics of Materials, we look up a table or a graph for the value of K corresponding toa given geometry. In Elasticity Theory, we will have a chance to derive the stress distributionaround the hole for certain geometries. For example, we will show that K = 3 in the limitof a � r (the plate is much larger than the size of the hole).

We now consider the more general case where the hole has an ellipical shape, as shown inthe left side of the diagram below. What is the stress concentration factor now? Althoughwe won’t cover it in this course, Elasticity Theory can provide analytic expressions to thestress concentration factor for arbitrary elliptical hole in the limit of an infinitely large plate.(The results can be obtained from Eshelby’s transformation inclusion theory.) Intuitively, wewould expect that the stress concentration factor would increase as the radius of curvatureat the tip of the elliptic hole decreases.

On the right side of the diagram above, we consider the limit when the minor radius ofthe elliptic hole goes to zero, while the major radius remains constant. In this limit, theelliptic hole becomes a slit-like crack. Intuitively, we would expect the stress concentrationfactor K to go to infinitely, i.e. the stress field would be singular at the crack tip. Indeed weshall use Elasticity Theory to show that the stress field near the crack tip scales as � ⇡ 1/

p

r,where r is the distance from the crack tip. Based on this finding, the linear elastic fracturemechanics (LEFM) has been developed that can predict the critical condition (i.e. stabilitycriterion) for the crack (i.e. whether or not the crack will grow under the applied load).Stress singularities like this will cause problems for numerical methods (such as FEM) if wedo not know the nature (i.e. the exponent) of the singularity. Knowing that the singularityis of the type r�1/2, more e�cient algorithms can be developed when solving for the crackstress fields numerically.

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6xx :Q ,y f fV 6yy=* , r ^+V ' 5xy= -h r y

Na- c^aA r^fu^nr* _,_

tLw rryh 6u t*rn- enld^far,"&nx f * y . y+ .Fx - -eCyr.x t 6yy,y 1_ F/, :O

a4^t6 rno,*',c^,1-!6, s r ;t-;1\e o/,uI

TK-L

fn plnrw s@

frd= -t#ln'v I n P(*u s(rvl

96. Exr'rwpls -LytL& t/rpttz rJ r1o hoq ft*, N,{ Aa^re Y*P:-oLn'U ?ie L -t-r;N( J00,4/4ffw

f C x , W ) c { r f p : A + YIt oFutouaQ- s a,lttfu th,t- u{nd,f{r$}t Vq:oDhA/t dOpZ -frh/tl So&,^"{-i\s4- vn'e-s''v\ ?

)-rtu! Frhd ov,,t +fu- St*s freta6x= f,yr : ?Qy: Q,r , =or/ hry =

Me 3Qo9 + L L z

-

L*L go tn lrgW x,tpr prywrnob

Qt r ,U) : *nx r+ f t s y 'T - c x !

5rr: cyy *

v+d :)

6xy =

--t

--).- <-

()" ?e-

) lr" f{,Jri'h. o,Lw\.rfl, tlr\LT-lrY t fec,fu A^teA^,t/ryi *,)/in[ tdu''{r

Qcr ,w)

fod1rsh,UrrtMn

f"

Strvd'yb"",s*w

l'4= - L;,b 6r,ry) \ ur - l\AJ- X X - - Lz

t - : t UrJA: #4 A

I'rrwm-prnf, "{

inprrh,NI

6xx N€t I

Yt I\ 1-xb z

fif;fu, ol'wrwt tl'\-A- S4'r:,/'rs ft"*--t1r a. te c/r'.tf,'uJa,t bv/L tt rtd-p-r

Pule ba'ntl;^t ML ^vtx,w ) :

v+f =

ME333B

ME333B

ME333B

ME333B

ME333B

ME333B

ME333B

ME333B

ME333B Rectangular Beam

ME333B Rectangular Beam

ME333B Rectangular Beam

−2

0

2

−1

0

1−5

0

5

x

φ

y −2

0

2

−1

0

1−5

0

5

x

σxx

y

−2

0

2

−1

0

1−1

−0.5

0

x

σyy

y −20

2

−1

0

1−2

0

2

x

σxy

y

ME333B Rectangular Beam

Mt3?70L/4@ Re,*l**t]

6l -W\rrnffi ,{"r ^ )

tr t*z;-"r" getr;el

* l:f;crtf5crt ar * 3J

Tkr_ ot+!w*ono,4ih,a 6 l

+ {: { r*s ?3'i''

Ci = *fi, fcx.,grcxr*{r-€{r,n"-vn\ *yW

Q' = + f : {cxs f ;cxt dx

ry N*ikt-tk fo',*"* ,tr a'9wP

J T l

Crrlr^J*, art an6;t'a*+ h^r*u,T If ,

ItxS a*(;w sw - q< t< a,

IA g!d,t!t^!, N€ seek ^ se*A** f^c,{'1w,4 ?tcx), t'=1, Z,

f"clt -fh^h

i.e ftuth eaol,

aru n{4,0dh!/v

wnd that lQratj {"*" ^ o't*,ftA b"*a o/u- dnrna-n Fq.(fn* 1wJ@wn ib m"* d;ft'"c*W * P*w )

t*(,Ty f cr: qa ha- aryw"d+,! by^p,4Aa- CL. an-a .UXyannn*atftrrtte-*r"

c{'4dt4i}t lttl.ys w .fnd +t + %yuna;svt04 '

r^e,d x 7 c *$$ ;c , l g i t t udTt actA

:- Z C^ Sir' ==. C;t-:l J r

F 4 8 3 3 3 # *e";eJ o"fiG

8a*^* F* =f - r f tlfi,ut i : |6, M76 t;^),x,

C{5 Arx, J,v\\-x,

hn: * r, (yr=r, ., .-.

fi e, fuJa"r1^ kz4eo .t . rE?,= t

'(frfn*x1

l

!,

{?,=c*SX

fu1 : A Y t tX ) : Iaea

t <i2+ L ana$)nx+Lnsn) ,qxn=l

, C, = Q, , Cz-: Lr ./ C3,o 4r, C*-br, , ' ' -

\-.O - f cxtg,c>ct dx = *Ji f,*, # ax6o= * f i +o.t 4x-( )*o\

fi,6: * ll f,r: c,rshnT-dxbn = *- H, f,x/ lnhx dr

1 = Q ,

trsyrrx : hc-Ji.

n:1 , L . . .

aa7

t \

t;?a

\-o "aI

€a^

r f aO" J-or

r O .t]*0*

I

1I"J-a[n

fip.{

7 AI-3*0

I

a}|I

a )

*Wt t : * " ]t o \ x - = ?x dx- * .

'.xs,/tx

r ).r,)t (

,\^,^"X

a/Y1

txo - -

C.tr>

' 5 1

t{r*f d,^b is

XJi, g, g6sg,* H go t:c,r

*J-l g' tx"r

SrtnWtilQ., llnnf t , t ) f u

q0.J r

o

:.o

J-l ra r t JW,

- l

^ 1 o l- l-o'

{ t j

r't le

= LI.orl-o '

l o \1 *c"

x,7t

be**I ^ 1 .)- v\/

Srn)"

o.{X

t An ,

) ln

itn

erl

iI!:iIiqt

t 'I:!i:t

* f_i c4r ln X Cvr \,,\x d,,X-' = * ffi c*S*+),*)x + cu>$* X*)x dx = 5n*

\ : 9in;;x r'x )*a otx /)1 /It:l, >/ "'t , ^ ^ /

- da[; 6$^-]n^1)x * c+t,]"rb]x dT: tn*j-?, *rn:C J,r^,xnr)t d{ : c

JA

J-a

ii) f6{ t>.1

9S Fw.e& @€*& b;'p,

4fe fo, ,4 a.,n Y f*'|'Lx- , *fot-fextfk^o.rt fr'-r +tz <txyenw;on f cxS * * * * #,a,ncs)ffx

-f l^ rn ffxb n . : g h - l r 2 , . . -

CX) - * * * Z , 'a^ * 'Wx ,'.f , *eV CAfuZ k/,/"40

fwft ^^tt'7

i-,a**A

4r

n=@-)n

6^tr &"> b ruot tu\n- -,% *t k qf^d frr, i,,/, ryAL,fe/'r^t"tt.u.,W, M caflt wvrts

U

fcfl = Z a^MT.nx.

= Ar cr;hx f

,So Ny ^^fV W fnrx;Ns c.u.n ,--A*,k" 4r{- Frl,nynr++ c^drry

* l-: f irL')!*{xt kL = * I!"*lnr6t) '^x o{x

t r A- l

2-a ) -o,

a: A n r n

@6n+ ?.*) * " co ft^-^l*),"ota" $y 2-= (*t'ft

2i6

zar ?trA, Caj>'J-7- + Qs 0$'--X f \ ' '-

Z A ' - ' ' L l / \

t , tUohW: f/'ttt'z iz na arw+n'r* +Ua^

( ! : t I' L / ,

9+. T*q Pr{fy*"qftq""k"" Lary;" t4Ja t* 6}s" 6,"r*v"at r l eh 32- and 97 , Nt

cn/n bu eXP ao'fudd

r fbat>att

tgr.'n,j * I{4;c^l : t

ho"^tV ge-e/n *kwf

rh ltt *jt J*nanz

b, *s**, 4+*, kt#^|, - 't. r r -nF . -e7T6* \ 4h^. , ^) f rx , - "

* If a'wr-p nb(wn ir or.,], ,'u,ftVu-d "njor(icSa- Lig cG'?\ ,i+ en'+kql

te-n'€,A/ "tryenne*yn.t ht{*t*. ",.9 ism o$( c,rrnU?znricaJt.

tv\*373 1 -t*6lrfr {t C*'

9t-

9 J ,

aJ-^ 92? C qW 9 J

orh'nn{- fl\^..s+ffh" Cerv&n;

a,/n Ary" f"<"W"*t, *awb A{lur^'f

ia^(44a

Q..

la anx- r\rrt 9u m"J#ft% "f *.

trn on-e- \ol3 \$4,' "n,^"t*ipio ,l A

Hui^r Ty ?! fo* *1 b6fl^ c^r({e^tr ?

it,nil do *r- fi^'D 11^lo vvbs Ja € *A S4^'-:

A. Yot>, tro,! o"u-6stl^ ocrtu.ty , but Wn'J o^z ^-b>' "t{i€"{'wiJ /

I txt= Z cg.wtb.r^ tf"rtql,L f*l a owf okf,h{d h th{ d,svnM; -n < x ( q

tlnz riahr ha,nd trsLt- CRHS) ;^ a'f'u"d in - e < x < @t J

f[t,u tuo Set\q&, rrlatot" ,\ fitp dxmyvho * QSXS ft , (o,rstbvU:hot'{d, f l

' /

b."F ,*?"r"n 3 3"+n ,mr*ntn- tfttl lx'mm; ,. - , ^-

i;'-"'- ?--4.

? S tr'd'^, Tr.rnsf,tt"n

fu f,.o'rxSo fcu

lcvt * j : [ fu,1

t -oJ 6 r -

Ixf ' r r l

Jf or - ,,,"tlxl

l5t ' ; ;a= fSg'Cr)

Je.n'ta a"J I

Far)4

1b

KLa

It

, ur tcr.r-aLFz

o T\-

L/\

ffix

3cx1 A ?ne-Jel*a{,-"r1rc" a*$wl aad C r ) = l o n l oouo d

ftcr , {cx tdx = *t"t)-* 'r

-li i sgnftl =;r'f ,ft )I L- l EI

t;k

u,rpo:+ n o a^t g{rr* o.n o.r latf,1-4a', &* {* fo, r'4 *oo < x < rJ I

W{- Cnrrl Sfrll te'y+>+Yt# r? att ^ f$wpo*t},?f,/t t' .!rn,a:s:da,{. f^r**A

+ invwS< Ftun^nTrwv'afa'm (nf)

+ Fvua',*-c 6qw*twr"

4 ^ , o r , r " l K ) (

+LK) cam bo reg*vfud al c*eft''r^wnfz 'h /Yn^t 'l e = alcx-tit

k cun YV atnhnu*'%

b ^ $^'"r*rv, , w( cllt ;t tlu fux&" |-'E/Y'<vW'

,f fr*," fu4 (*{ sryl f fft !* rqf,h"*reorDncul c pac-t

l t

, ft^t: f|{*r)cornm v'n- fri*";t , fgr,4{iu,*\

r f ,{tx t fct( )

Jtx.r

"xlxl

fc*t= k I:: f ,xt, "o'* K

T IT+ f tp- ) )

:

#^4

frc+,

A -+,fu- F.ti-for\ h.<l^^atm4v,, ...| lm),) - )

;1n i rr I ;t 4,L Fuilo4 tra"n^tw"m +': t \ . . J . . . . , . , .

{*LR- c/h/r^va/r}iteE on hlh g;l*tt

l 'rx1=' # f: f t*, ik €'K*dxr l -oo

*t .1[ ( r r - t - T ' [ { * t ; ( 1t . n / - v - L ) - ' J

.^rK f ut * TL fcn1

SY f-""4*$ucx) :Jcr . ,

$rrxt= W

EX6^'"$ -

$in* )- zh,rx )

kcat*r fr[Scx,,J= t-' ' ' ' ' T *rynun= #

9r.

3s& *'4W Sa}*h"Wt,

pej4a'J ,.{ Po h *,r-^ol raed".o},J /

Fb^ thz'Jn frctr?.", V?f : o' €{* ; Qs yy €iv: fr^,

tya<.h'on- bou-r&+t*+ a^r'^&-.1;6:' , qo J n ' : 3 j

trt"! sgt^*ry' + tx*A, = Z cu fl:(x,fl )

l v

Wltotu {-rao$Iiwt- f7 o're pol,yo'nrr:o!. {"nU*-, it vnaLus ltznl*

h l,r'f f;cx,[\ Le f"Lyrurru,v-( ,c','t'*;rva

Th- b;Ao,r,o,aa'E- eEr"t*lian ( Vf/: o) a'r'ta( e acL B- C,

q,^/e a 9?h rf. l;rtzm' a$afu^'h e(uo'-fiorw frr{ a'e'{fi'a'a^i5' C,'-J J I( J J

t-'Al-*:?'n-J *{ f" T-sn";af ltten*d

l: Mot ci{4y ^'iL*t ft aa ;t' f c. Xl. *" noE fr?**^X f"^,4t*z" *7n f? t Yr*y$tk 3. xt*) arrc poorl L+t'red no'tmu*'c^l,3. NAa he ru;rs. ;/ ea"l, 7;cx,J) "!4%d4 S^+r41va frf,.-o

J d lS" tAo* B. C.s m-u ttu- *9 4,*f h ^"ry ab6?^f

Prowisz o{ F,>an-ten Mefria a(J

t. A /w^plpn/ -f-,*;//4 * nh*'n'a k Vrtf*'o

Z. Sn*t, € c,Pv Sol/v*:lvtL Sc'rt{\|&A V:*Tr:'q ;t- hsap\src^{- ry4e4*'t, 41n "^r"*f*ds tr a ?A*z.n"' &e +'ao;t'*t

3. Sy+*l;*+6W +t't tU S6{"^*-SnL5 ti ),rto^*d". +Lrt B-C .{ t:nt+;utat-J I ' t ' ' ' - ' ' J

t'LG3

?2, C*"t*-?- S&'^tw,\ bCtu''

^ t

l9{hs'n rwvat' @nevt$nIL \

Plo -e- g+raaln 1vvHa,r* ,*t

sw{t.t,

i L , n L -o(" + P- :o ,. . \

f , t ,D= r i l x e i rY

fg*+vt,sft!*Ju d,rvY\!,^',^l - , r t . l- ^ < x * a , - b s Y = ,

a > b6ry*i {{, t-b) = -ty{x ) _a(X(nny (X, L|TS: tlca(xS

s{ -. * f p

Notn- . An^l A^r"^,t"-xr"" co'r., V<- dc-o,rnn56p6 tnh &'wuAn'/ \ J

Y { r* , :7 :TYt [s ;n] r , ,x. : 7 c*

"i inr cn" crvylerx n r,rtbeo

et}^x:- c,o1 lnx + i sin).',>c

... tuxywrZ,n*r'nQ- a^& Jm,vtSoldJ f^-*1.-,r"r anl- ,c(oSe! fgtaihe

o\r b7. lrnhnL, ita icna3ryryq 0 *{

HUn * we, s€oK ge""*t*{ gol'rxtrrslrl'J & V++-.'c)

"fel- 4f@ gmusq.ids'{ t'1/14P,rx,V l = eu& gF Y

Y\ei&*'{ th*- harms-'n^'t / L"Pl"'*F- Sr*ibut'ry, W uL +'\ol- * t ' ['-\ ,

u A r /

*n,i*it}t \tf =o lo* halmo1ryrc. t""r,i.Tn rnwr'!*be- 6W: o- b; hnc'nstro" fu"'&tit{T{ )

v'+ : &* +Y e*terY) = F*F") [*+.eYJ:

yj q lrarmnrTic 'f,t;rrd"rXlt

M>\ssb!Ii r \ - l1 ftu,a;nn Sohrfffvr Cw'

r 1|6rf €aUn A,

:ihx r v , ^ ih{ €A I Wft- *r\Nt1 EoLArt"frJi14 jh pr {.*i{V-tr pLl A/y ' ld g ' V WIY VNv >UwY1v' -J t t | |\ _ v f

t3 4u'!. l.{a;vmxxvru et{i'f^*isvl vzf -*c ( e^J* tA zn{

$gca*'At- t& bi:he(r*rv1^-(- <qutahs-t y+$ : o ;4 q'{h o-,rd*r,-fzWa e-f De.f- +d,ul So!;v*vvt& ;'a eaclv e"'lfg'4

l t

",/hv,'9'14/,r e;AX ,J d^V

vt { a;^* y eo^i ) = t zv. e')' exAYv+( L| ; .^ Y e*^Y) : f zA. v- (e, i " (o^r) : o

Her"- +h.a- afrl^N A^N Sotrz*iow| h +ttz b; l,tar trunwiL zqualffnr A

- \ . v l V , ' l t r

g iAx V <-^Y a,n,J ei4^ y e-n/' v .

Sa L,',rtiu-u,- $ 4hL b; h a rl1..ryvnY .Q$UAf1nY*

w4/r++an t*3Qcz.U ) : u i ) 'xL&,1c, ! ) d l + (cs*c+v)

C1 , Cz. Ct, Cv

)rx" y): , c irslx I CA+By ) dY + C L+D\) e-nY1

CrJ& r.,r"c; + r{,}) : syr}"x f c*+ By; e}I t C c+ Dy.) e-Ay l, \ or\ / + {Af ginh }V -- €IY-.eAYc;rsvt iY:-:_,| , j t

, z-

ths-gyyl*,s9t1i'1rs't ?"^ ^I4o b€- Y,Fk^i Mf r t . \ | , , . " I(ww,,hr.l *rx,"i> - o63;x I (N+$l n:rhay + (c'+d/)JnrhlY i- - . J ' L ' _

- ^ l i I 4 t )) ( '1 , t "Y)v T

AttRn^s,hr.19., Ne c-a./^ vM- le^l* {mu.vvw e a,eJl-,'u'o"rFaa

,I

rye"^ t"y)

(.rJ$hx-r $cx,'t) = .rrnlr ltn'*Uy,t c,*h\t' + L*etPs,,"h)/1

Can bg

e-^v Jar<* orrryU.lX

Ltern* q.e&\ *€r,sn *aa a d*+V.*- usV*/o,&& ry**e*' il\ * *e 'f

e3+c L t""*"rrrU / i \

LCT4

S\"',nvv\afi(-. WV Slrr, 6t) filr[ , WL, rnle h".,rru

"(rt*;qd ^ su;!^s * jutuzr^-Q- s"1r^1-q'6^M tb +{ r $her"rqlrv1;6egra*nr^-- 6b+ u,^ l*L $&\W e,uw(tyrnrnWt) Modd ( A,\A;-s1+nmvt-rr ) irr a_ a,nd7,er j W q^fl&.

Qtx ,ut ): u,6>AX ( nfc"oLl,y )- 6 y sfn h XT )5A+g F- W{ -=;,e CI/f\ L ('J a/ e..tg,rn {"-r.^,{"\,\ {"r h+l- e tu,d

f: l. etc"wwLru If ^ pfft= fc,ct5%

,/' fr,1l-ifr? rb, F

--er o a.

Sec.rnTsl{- y&urrw rhto d"-tk^c^/r\rwrnA,hil,'o

+

af/\. l" fl-

odd in I-

6yJ a,g^,n 3 , 9r.rt,v\ i,r X_

&yQr

Qcr.,1l

L+n ),: h'", PCf' ) : ?, w Lrc = --ur, (t,!=b)

f b, C,, rnqh\afu4 a Forvr^'n,.^ +*%t J"L**mn

Sobn*Tsrn (6;g iairn't (al

tpcr,SSttt**tr'^t%-

Qy -- odd ,n J,%"sfrr- -

* , , , ; ) -

nnt"t jC*t*u* ',

+= ffiLz*( B\ unh\ t C h^h)'y,) r f - r . t 1 rY Lr ' - ) , , i1 4| ' . \ 0*,^ AJ+oclthhr

i1

ii I - t f r , . ^ r r I t . , -

i LVlt yt'rw n\4 d sbh^*i\tt (a i.

fu,-t = 2Y- 1L-?r * . - -L '^OO. ( Uf CI 'JhAy t C; I^LAJ,$ . c . d b g : ! p t x . t s - i p " c * L x j : b

tEy: *_p(L := t ?, c*)x l : - b-t (S A c.c>lvlb + C-! i^, / t xb ) : +? .

exx. :+ = u)lx t B; f ! cEl,,Iy f FgA+ c L-)smA Iy lB'C, fxx-= o t ' -Ltx Cstrunt ? 'c ' I

orrtn rn^{, ry so-}r},fiv I fu t/,re_ +rv.( qdrotr,,- ,l

Ory:-& : l-rinky: I C Brcht 6L^J +Bilr.rnh ^J]

B.C. 6a3-o Y=b (w,*at+^sl"* t B-C.)

(nt ch ) c*v..xa + fi, ,l-b sm)"b - Q

nL fr,\n <efr:frvrv )n i,hz W ,*n /e so{,wd AYPrtl".(* h /+rdC,"e-ffi"rrda B o,nd C.

tntwdu"r-e- sA,o* AB^d ho4a#fr:t , caL 2b = c, srrvAlt = J.

l D b c * C - s = ? o1

r v z \ t

I ( tS+C)" ) , +b )b .s :enrt no*i'u- oohiU - l*ri ^,[ * |

),1, coLAb snU ) b

I l ' , i , I / f ' ' 1 r(

2'.,,\' lL 4L Ac s,r''A )L

4

*-- . ' 't

iI

, -149i Lho i fuw;u" -':i,a{,t^}€rr* i--L*I-

"-I

i

/ - f iLa,n C 'i r ' : f t ^ , / I a | - . t Ji

'>'o (,arFiorr rib.l oa/A be fwn.[L 5 orrnta^lr'1- .|

\e '€' ' ( Jii n ^i l J ' L . t t - l -l r l\ = o

J

\= -b

[*=L^

t=+C,r',

-ffuI SsC-"*en'h tJ^b Mjd,\"f tN* \'i "G7n-;rwdSr^,yt*^,rty,sr2 Sttn'tirt Co r rtrd S"L^iibn (61

fetn &',, jlo L^;+r'r"

Frlr &y tunfrtR t 7ntt- "ftil,"s- Ja""t- tvtr*l ^ry L*Yh ttJ- '01,\/,A'.

f u t= +QH+p?r t )

l^^t unrfLl "tu [,,,t: L fufutu* tn|',;t* fl"A 5lnnO'ry '

+kfl4/!,,'t sgttn*ty k' 9- " d d rnt , ov'?rn ln fi )h a Jivrn(An' *W

lradr'n* h\.h Frrrru;," nlatQla&

{cxl:- Z a,^ w^,,}:, A:l/ \ n -

(zna ^Ww^rA )Gn4>r

zq/ fl'rltn- t s oo n'-o \( -t<,r"^ /

L" d,*l-{,4^N,Y,trctt*r"a u,t\tL- 6

(_A/A be_ trhytd

ktar"nagI

a , n = + ! : { r u c o s ) r r , &Trial- SoC,ohvn - , / odd in ! g,x4n l,tX

Qcr,U): Z, b^ularsh)^! + C*J;nlJ^y) ryLtL-,, A: I

%y - f **: E (%! d:lJ,! + Gr;"h )ny) (-^il crtA^x

ra1 - e yy :f=,t Bill ! cot )^! + (26il,n+ C,)I)-r;nA)nA ) niln xtrtyrz f,*y= E,({O"t ot)n)cott ^^! { Bil^ ! s,^n l,)^y ) ^ Iin)ny

Coi

/+" a,tL;;tray b"t;*6' o'v\ A Syrnner','cstvrt4u^e 1r.9. ^ Yevt-61/, ) L^'vt-

^/^*tr be e,v'urn1vy-4 ,'rh typvr,+

+ l,"a,lx'ry' +L,srr ,^e e, Il/'!rL q)e'vL

'{ oJA in. X a-{ou Y.

SuJ nrneff ial,

tg : {ven in:(, oJd i " Y

+, ewn inr, ?dJ i^ 3

Lo o.d" n *fr'-\ U

I. / ' r / \ i f -Jtli@1qL j Lo-t

tl

/ 1----r'-

t

6 o*"d cry. cr-rtd^-+fBy\o ,afxX-o, X: tot ( i * - ^/N"tu-mo./.'c^.t/"4, Mh"yhTA" d - l

fuc ,ru* b)nA: o ,aZ A*us^^T

n-lf 1y(x , ! :b) - f * , ==IL t k y ( x , y : b ) : o

r n I- ) J t ) n b C ' D l , r ) n b + C n f i n l i d \ ^ b _ #I

| (u, t h A") c,$A )n b + tv,, ^n b thA Anbft f i= / , 1,J, , . - -

- o

m,.Oolg 1)- f ,7 t 7 , , ' -{.* ery

t Nol',r* t{10,,t tAt orfw,/* uzu!"ai nNf lJr k |f {cx; r--s gX /Md-p/Irt thz " \LsMi, o fr^/?ti^ gni'u.F O

kron^lu co) nT - t / +a1. ea Ll\- /'v&a "U- do-ez r\,o+

so+a\tr tlg t"f"v B-C 6^114. x4^ irtdAvtrfu&

1,4F3333 -|*g-*o2 t . Prob lqrra S{'e*er*e*:t

scr'1gos-*-. th*--tT sw{.r.* {{:o) * <b*" ela,afip h^!lt-?crcs Ciisol;^ s*\",red th a.iv1 ar*{d{-s*e l"rdts drsilb*tre.. lyt;-s,w{.^* ruilt be- fu #trtar*r^*q,'t $efd (shye c/,rargrJ .J 4,hL

t-uq Sn.face ? UyLx, U:ol 7 6rr, , U*txl!-o) =.Qcr-twhur wi\l f,g +he

v\cx.r

/,1/tqrlvr*&'l"{f ry"-,}s6

VownA*t rg, Crrt d"'+sn,

J '

I l,{.94/1v

6yy ( ic, J:o) = - fyCxt6ry Cz, g* ) = o

7hn ezy{a "ryd btr 6*"P, Gad+ng coa be uo'-61f+z"ttut a.n t*fyra*{- Ah* Qwy"z-S Grc*^ {'*,"+*w,

Ow ga"!. b 6.,qt44t* tla o *t',1$t sxf'wffirn * * ,, I ' - .6 i" r*s Glatx,gS

l,S ttr-*Z < o

4'eq41vs- Swrfars 6rvrn, fr^r*-* 61,, Ct - T ) , /

d,A /auz,^a,^,t" ',h *th dtWr*rin qn -f hz s,$a* a*yoUk *,$:o)dre tr & ^,r.a+ .{-<rc* <rn J\+h d^\€c*iv?t ,- +r* s^"f"*- o*-yoW .t{(y:o

fyat = f:: - fytx,t q; (x-x') ctx'

f*rxt = I: -fycx,t aiy (x*t) 4x,

Slr.l/!t +reld, daplt wr*b*k fie\& rrf +t-r- <^ruUtt- hq tpuo?611 tx , $ ), t,/.'4 (x,!), 44 cx, ! )

t=r--{*tt

ll",^r"'Uw, s6+a;dt\fi thv S*4"u 6rtr^ {"'n'aw trt Yelncnl c/acz-b eqb^*^* 6"odn+id*;,^tr a Sn"+Ad.a! b"d an -ths t,-t'**

6yJ (x, t*v) :*.g lx, Y:o ) =

(7lazr*- Sia,rJ'.o)

* c . f 1 l , r ( A + A 1 l

*W eLj t*rna b"^rtsi^iu"/\t '{h4- t o{arCtwu h

M*7333; fu-y-9z- Sr'uwgtrdJ" Lr**k D-vl I*tct.{-

TLe.rrg- tua-Tna/vvJ r,ii1A-Uo dttai.un *g S*ta.e- Gyqrt/rf-'rcl"-u-"'

0W- wo'1 n\ 5 J+v,r'l- o{r*"{"* SZ Ffi^ntsx" {rrg1r^r&J.ry1,yi (1,4, r'4 fe*,-yroUlS,

6; ru = f :: oi'crt e-ie dryw '^Je od+a& 4,;rlr, , \u ca:n st*ai,n &*re*{-efece- saL*av- u,y

inWz''z$ p0.1^'|,-lx frdl^/1ft1,44"

6'yt*s - !r J: f; ckr girx n*

B. C.

Tyr*,[ SoL*rtti:l^,

Q c x , g )* We- c,fn o'vl'q-

beca ,t&e- rl'lg

€*-= Q, vyTy = *,**6xy : * frry

=U:O

r.f :o

S*g

"yy

a* U.:"

6il c"r

@r'ndrJv lr.Lxs * f" c,*Ax

fl" S{7Y14- wrfiY-*d x

taltu,^ofi**U * f,-4^* /++"*W {n"&,\)- IW6) : -fo cstLr'

( f',clirrn{ul, nn +"^gv"+-t- forco I

e ly / n \

tA> ",)tgl,",a,Aer" 1* **.

n{rf +hL {xYYPnnwn fl)vih,

6tur* {{nrnzsl) +e*yJ d\Y* f c,si 7x {A+E Y } eAJ\1,\1,y [ 0l+B ) f E?Y] e^Y

h{Aa+g l s ,u , }x :o - * B : -Av-A x'tfrtxx = *f.r,l,Ax -' A: +

6x.,x-:

€uv =t*y =

- f" 63A?L (ft ^'J) e"^1,- T o* !"V sin.i \ dl

:;! ct - <-**+J;;I

tIii

Tks- S+ra't, aun d. d,itytoico,w**t frt & c^"^ be eaA.tg {-,^^ d

6e**z*g"F; il*k":, /*""t * f* 5+Ya"'/Y3

Wy rolt:a -6 v6+a'i'^ d^>ylac'a'/{'I4-(''j/"4- fi o td

rtr*a,y),= - fu sh\),f, f (Fv ->v") t c ltv))y i

u^!(r,y) "= - # cxrhL i (z-Lu-) - c I +D yJ

pla,rnq g+rd*"

fz*: "tSr*+qt)I Y- +V ?" ut\X ?' ,

e^ f+ c\

e)v +D. \\

r;a7il-boJytta"voa {C,to+rsfi,eeJ-n eui'

Reca,r,l tfoef th* apyt;e& L^&g' a ?y cx" = ?' c'$VL

dr* Svv$* d,ay{o;unnwk un wytnu* a

ii,txt: * *# ?y&s (A',

€r": : - gcs>v/- [Cr-u- zv*) +(t+r)AJ) e' \v

QV = ecarl),x I C r* | *tv-) * ( t+r) )yl eAln l r lt*y = - ?rlf snax (l+r) y u'/

ffit*, : -* einlzc (l*u*rr")

{rtx; * -k cir}x. (>*>v')

* 464Y4 #{r"* dt* Swr{z-<x- dap{acer*{,*rt* Aan{ 'z,u*rff i^t"r*rc''rfL * A'

hap,ut t^f th +tot k^d ?vul,t

Y4*7V3b

tAVe?A\ f,tA'{4-1514 i'n,4..

Qi1, ck ) =

ff fi (d)= t;'nkx* ,t'/ i

r | ,/-\\ l / r \ x-\rF1a-=f--+z'I

l lt l

Llw. ur +fur- f"4,V,Jo ltn^,*.rr

Nha*

g"n'a tha*

Tgcx; : c'askx'c k.> o)

,'{ Iocx t= cvtkx) ' t

Tt^R-"k\t<, rwufi* t&.f/t w *rt6.* _t-W& Vy iry.*"* { +'t'* 5'r li^,r"s,Ve capp cJ rpacU

l {z {rv') _L

l.J< I

ytotu Co>ka: c<S!kn) , l,e. [aal b uua'rl+6/r\N t ke -k

tt*rt 7,:r2 --->- th",t {rat= _&p *U.

but k<o ?

so fl<* J">faarnurzf a o-W wetLa,N-fr*t*--> *h"en t u4', | -fyo,:- "g fuw**

dr&uu- *'- X- T.t z k

flLuv $y(x(t* solPx/

W J- askx'4 | L , C Z ' ) = - -t EE IKIuyLxs = - ry , { , srn icx

h t+t'rr* "j Swltxt* Grge^ f^^*n, un-cav\ alrta a/l,ke

4os * t:: -f1rxs Grlrcx.{s dt= JS - *il*' Gi, ( x*{ ) dx':= {5 eK* gtkx't Gs* Lx" s dx"

= -{ t: *"rv*t Gficx's *{r} ttLxL J - " . " ) I

€fr,rr= ggl*,

x hA,a WrW is'(rvu- relw&Wf lcz" '8/ J< < O.

b* x-{-{I/=X* 7J'

ds! : * d&'/

t^ n'L

$ ?yu l= €ow : co>kx -Fr 'Jh kL ,-?, \ : t - , -nh* , , ,L f i txs=- ry fu( " r *

/\ q r'FX: = - G : . , C P - e

Ii rnr:33 S

bt 54"*,Grqw fu/r*fin i'n

ryQe*f Cpo"r"r-

"te{tf^rf+is1,

7rE

e FYJr,r 7y{xs: s;\ak*-

= cctk/ ---7 ih!/h

x'=X* &^ -n l.ply {il = eun- = h>kx +i t{hk*,

-----*-+ t/*'1"1/\ [' .,-Zil) iG-,,(k) - - Lr:z--",*1 -',' ' E K

C*tr*r"&:uz.ke& {. ^61;np

ul^-r,/r1 +h* s*Pw U^%

?:1wt- ^ [w)flrl, qtL su^fr* d'aplac*'rxa'"'* t'a

ftrxr- G,f, ntfry ct t = Ql, cxs

{tr" mf w +dd-c-{x,ft*Y-n rQ,"tion*lr lxruea arr.,d-f-s

ike,u i*cx-t* -Q-vt-v') * s;nkxE K

Gfy cx:= * zQ4'l I^A"lul

6fyc*;: T-' I f;i, ck)) : 4* T" t fr;: 4y (-t t"yl'l )

vnticsihat p:2)^U+/)t"n plano. gttai,n K":3-+/ , T+l: + (t-y)

"Lw:ryw*g:r.rl.+/Gsrrcx)-* f f i Wl* l

6, , recajl {kai-

?rcxl= (s)ktr ,, ----2

(*r"x- reyt&tnn "t' Xr" 6-/'

Xx(x,):-U:!#ng snrlTra)= !a€12 f tc.r tkx

- f ' r t ^ r l Z \

ff* trt = v'u-Y-) f ft,vrkr+ {corw)= (t-t)1p. i oiro

E 1 { -

s o $

er4tx)= - A q€{}?Jtil = -!M}. lpntxs

Ua+, -+J->-?.*+:c

(Il<o.E)

Q + Tuy4tt! t-.rad;,y 'n ' 9'nfer .

W e t6vnit$4/ o??* a'o,tn, v$e ca'lr' s,l\nr *fuvr

f1&? ey*a

t:!-t: {1.uZ{t*ru}* r}P- -X!L* *${{+v} +}^ 6r"^

9S Atb;+w,,6 Lo^d Dat*kxm'r

{ GLrk)=Khl fu|

'' --)

6rt"clc): ffi t

lh6&) *

Ut {L) =I

U y { x ) :

11w; ruwno'{ t,,aodfly

ry

QLrwr* # t ln [x r

Q f , r x t= -WWlL l6yrr*,: -W- sgn (x-)

f: -?.1t/t t ffi 41"-*'ll d*'fS -qo) t # sgn 6-x:) J d*(

+a/y4^n&^,{ {n d*k

{ a u 1 :

MEJ?}bQ 6 Sw4n,*," J a

Qrt* F*,r'l "*rtn ,\^ ?D #avt*si ncSX yntl*"tf

iln *f**

r:wR:mF-r*]i-EA

Thz daS>Lwn,wtt,{tt ld ieL*'lwvu|l

f i r wl t * :

ui=U z '

bry

htz! , , r*+ ( t iv)9,"J

# f- 29,!z t (t asts I,y)

*r I z f,*a a(t-v) 9' * J

e&\ard? vt '/h* g*&at* ( ?:o) i9Tt\e. d;ogta( f * :l n| ?\/I uy=

{ * =

. ( d "=,1 : *I U u = '\ 4 =

ffi 9,x _+ jf ' :ff iy,rffi 9,t t ffr*o-Wg,u

t-->v F- -ry ;nrt-u F>

J^ z]{f

U**V e*dr*wi6x-*:6y*:o (*:" )ct*= *F s&,,r6s (*:" )

t ^/ f cK) fr-L&,$)'j cRr ' clt* ^ F

l' o-i & #€'j 4 t,L l , 4 t

ftfr.s ,* _r_Rz

urie./,-s- q= -S t"A@_*l=_-E-gmF*_.)

\.9:-# h{*}"- aJ) q t r l ! -5f = -'1i trt R-79r trE?lu-" - - frra q t r l _ l l _ r );; = - 2rr ?-e'[ n -' Ia q l F. # l

) ? l * = o . f T I ftr

ME3J}

{3p)

(3=o )

i,,*U,W ) =

4r*"' 't> *

ll+{X,W) *

L- d/(!A- **4, tks V? s"""J"'* Gttve" {*** *

. I - V IQ*cr,,t)= W ffis

Q)urx,b): J*zv xw-w6t4*cx,U): #, h

{,))**'4

Nhnt Athe s,.,'{au' &'ayluwnl ':tf F"td /^}'4 t'

tuJf+w"g ruv*nnl' {,0^d eo1 eu"J*tr"2

3v6,V)

r4t733 Spao*

6r**n &* r*ts,'r. t* 3B?e.s"\t&#w

fJW* N* de{;re rdbe-*-*r*"u'rt"{ n.^"a^ 4w*'04<wn

fur a $r*'*rue fcf, ff"J otel'c*;. #vt * aQ < x {,&, *6# < I <*,

*s 7T ftuvt*tt +Yn//ufs'-4 La

?or*, ry) *fl:: lrr,u/) n'e*-iry w d"etTh* tnVWr* Fuu;e/\/ +ra^nn{a"tt /'J

f,,,w, - #y f:: I:: f rr*.W) e;txx+iQl ar^eV

4 - +,Lcr,,ry) : Vrr[f rx, rtrs7 fr^,t): f :rr lrru"q)J

Onvtrrwft- Lp fvr^t fe^ TyM\A4"&tw\ P^;fS

f tYr, arsfw, g-t

Qe4w

€Sr,W)= tcxt Er V,IL

r ; ryFzTd- -

,^_\f =lx?,,1") |. zlrr-*nt"g"

&t-#WY)= -#r-

&t-*Wv)=-#

1_Q.rrf 4{r.4)

e-dl+#)+ @*{wr':E

iFx*rytYvG

C-;

flta a' 3v hftsfn* a^$'evted 6n,r'-*"{" Uriry lze,Wt,

fu S*4"^{- dapto,cr*e""4' flc tJ c"* bt '^ru+k^'\ ttJ*l { +1." I'^.*"-{- 6rep^ $'*t'l-r-,

[n tx ,V) *

ty tt,U ) :

{ , l * , ' 61 :

T-^/, -Uh* SDecA'^JI

f rH,W) *J l , - NurLvvL U7(X,W) *

Uy (x, t ) :C:W lx"u]) :

2 sQ u rt -

A sttl zz

c*hfi{7r{y

Il"" t: - l*atvl) 6;rff ff *lnt*'r,t ci"

tx-x',9-Y,) ir'q{x.-x/, T_ y,r Jx*y{x-4, y-},ld:r'dy,\:: J:: *{*tx',1) qL

6. po',uut f""" h"U?

= -Ern Cx,!1 =-[ tx > 6( y)

(^+ +t"y shr*!d)

c",^r- $. a Sinr't(q'dr.{ h"dtry-

etKrx +;$ Y4 q

VrZ

2 s7lz? sstPlt

I* flhl.- tpu.^-*.{- ca;!{- "fa/ ' * ton, r ' . { .

116, \ l

the,n i* r* ,U\ : q j, Cx ,y IX, u,,1> = 6rlz 'x,Y,4 cx,l) = 6$, cx,t1l

(12,ry)Ckx, lJ )C t n 4 )

f* { 6fy*c,,vilfr, t 6!u cr,yil

gz 'KrX +tK1Y

giVxx +iYvY

gi?xXt; l* /

6'f"ene- 2L , k,, fu) : T* t ct* rnwT,F^, \ )

( v<. ry)

fwe" A"* f 9=n , w Yeo&ltt r/,p >p ?l* slra* /-1"l*

)

;tIt.48333b i1"&-1rc*

tAv(x,V,*) *- (A,t B,v) gi l<*x*i?' ,rnut(y [:r, \,2 ) : ( ft ' t$tv g' ' lc:e XI\VYY LY*e

u\" (x , 1 ,v) : IA l tBr?) g i lcx{+ i \Y "w7

fdo te thah tL.,D\ tnt"{ foL,t*iu.n dou nnt

IL{-IJ,4 oat g S*titfu tW e14r#"rbt1'rr^ c,o',rd.ttn

w)Z h*re- 6 arnkdrs^xl14 (ft, ,$,, ft>, 9t, ftt, $l)

a"n& 9iy eXr,v#-l-atw: , M f*^ r* (c{/\Alrbv}\"'*,-.f:xr,*'wx

l'IJItt

Cu'4*"{g.o-* A e\s W canJ- t tilt- h,6Ncor*;dep ort e{a*4t( rtq ,f?rer{Ud"€d b rrnwg"sr&aA y*'N** s-*thp-

W Sr^1fu.ea.

B-c ' {q * tx , \ , * * * ) :o1 61t(x,!, z'-..) = oI sr, (x,b, zo) I * (*U'\) =t

fqrrti

Nt txyert tht- sot""tium ^rv?^Id +",4L fkl- 1"4z.4 "f , u.3.fr7u't) = f i c k' , 3) e'kx+f rrY

csa&rd.ir|l,

c6y1d"rg$^

*," A>,

,@, A *tu,W rAb olnlrrr& pwfit"w,, ,u-ww,,,{d cr/.$;nd,* Fr-;.^ W^'J A* 4",* 6rur,'f*'r^mw,

W€- hortg w\€/^rtit^d thad- 4n 39 t* tt rnn& C.u-ltwn'trn#' h h**

(/tlg-U*0x. wi{l,v J,q:tacerr^na"n-,t .frr|d [,'tx,\,?),, So f/rni*dNnl hqre- t" w"Y. a,b-n"* v{.a- o"Yo,'ttbiftq 6'\di"'+r:n

Mnh\afted 4 fnO 2n eet^6srr, u"s urs*$d Gla tn hnp *"iltrurydtrptocer*alrt' f.etJ $^^.,r"-*'

{^tfi..&u-h.:J*;ry

8; -{3s

: r . A ', l4^W"*..:_tr_

lln =v'& v4 k3

) tu u -I

IAZ =

I ifx1'l 'X

ztt Yt

Ii\IIII

L*rt-tfn

II1IIt

The e>r,lo;*^*nf, 6'L+hs- t^P {z-s) L4

W cr,ff ) ==

iyc*,W) =frcx, I I =

, & kanil 4^e-

6l* ,r*, k/)

6L ctrc, ))

Y f,^{"fwb(ateA syaen"

Q l n t * . t ) :

cir (x,'g ) :

6i* (x'u/) :

l->vz)A

I- }V*qt - l/^

I t r ' r ) + Kzzf e! 'z

* + f t t*rvt+v"z7skz- +, t [-" t-v] +Yzzl sYzz

, ik r+ iWY

giYxx + iqt'

{; vryt tqY

ib. . i kxX+ tVrY*77 v

. ! )? . o iK* 'T+tkTYkzn

v

. I o t \ r x + i q YKz \/

f,una^ "f U" "f* 6r*n

l* zV it*-,/.1 kf+ki

F>v iw* 6 Wt-v I- T W

x. F

Y,. L

Y*t. T

fi"ten fh P.I , arvtlr"n +1^L ir-PL GNeu 6^^^fifr1

t-Iyq{r/,

r-rl$vJ'1- t)24,jA

Gn'd,tna$t*

9t . Tlts Ret,^t -i'n- bdt^eo* 5+ceaa Fytd q^d -.I+re a fu4,1f1:

lkfre a,re ma,'ru& rrrof{*r*a in. ( j r -

h d"is oun+ polwrU I

C,o"fr,dnna.U* Cx, [).

W gt@t$Vre +hr*' an** rnme r;rn*r:'iq'E**sa,o-Ldi^0^i6* (Y, r9; t"n*end o$ c"rl-*,an *rt

J

I f: Iffir-t ,q = arrc+n n Ix

6)L Ra^,eA t , > L

*y : # Qcx,V), 6ny = -ryQor,ys6?r, w,, *g ia f'eruma tJ' gcnol

Ln C a* qel,-dx &a,?.1axx_ +f cx.f i ) .

71i4 rl"cnrt wrrh & Wyrvvt

d , ( r : eu ? wlr"rz6fr^L,rg i, j : l, Z

' ti = 6rf 6r= 6xx6z'z: dert- Ozr*dy y' €,L *- {re €,t *6xy

{ KF r cdto[ $ : rt;hs

@c : @'g )

fhr;a ^"sa-{d "lhril ua h expfuo t{;'' q,g, €OAnn J y,XK, ytt l l , y,F/.

- ' . 6 t ,

8* d4p?f fu wfptxa*L'X. ^rhj"4* u*g ru"'l1tt1 /,*tentd,: vUIM *r,6r&,6e* irt +4/,/nuJ 'f Qrr, ,{rf ,I "

.t t fTh.B c^m b+ dwru- bg u.,h'ty- tl.o- cAarin ""Jn fv u

##+##= *n*

{= 9, fu=9 fa,,l=l^n she%*g* %=gl t ')r

f l ,r,c 69

u,tu r"rryld .6k4- b-A

foCI

/"'V"{yt* a:z*z*tvU,(

. 'M?AY b 8

("ff'' , P*htr,o) =I o 4 - 't(I kQ{,,ss=*Wr#6= s* r&$+ry#

Co-ruh'Mle- o'vr wl-t/n tlto I'u,a'rd fu.r*a+ta*, e.q,

4- 3 (+Yi -cxro+fzst- ' * 'g f311)lg - fr [a,,r-,r ot \5f )- r F LAI/

^\ / nr Sftg aQ \ S$ns=@e*(* '*--r- ; t r i - +*("n* +q#)?- -3J. )= 6s ?% + sSE i+ * + -L=?-l Q t rsmsco:oi*-+Aod f ' f A r ' y ' y€r * 1 1 )€- r 4 a o Y

U +A"L trd , we caln e)flwvt Qr, fro, 0-st9 ih +e/uma "t Q,r, ,Q,ro ,*,

An o$te,l*ntrfl.ft'kg -flrzu'*"J- t{,va''f; Wi{r( wuaw off ire fwn*'f*"r'bl' U b,--.--*

v\4rvd.l;9s tlT P, gna,Wt Taw+r){ Y , "0,nrl" N a!'ao a- t/v.drt.

Fvr ao-r dv,b-;W f-""*u.n {Cr,|l,

Y f = * * f + 9 u & t^rvfuftoc-e. f^^W f il at'br+ct"t$ ) unj{' u'/vl L

r : % & + g STf.\^) CItrrl*,rhr, haa rn*+,rtrup ^t '+ it 'oy y1aa/ ot lsw-fo**fThr^tj"6"t" u)sc{Jt V, *, & Tt/yo*t'rs.

Y- ,s a-.Vestor , bewuw vl{ eqrl sh.q:d fh^i: it sa*r:fua

#\g- fua/t\hf"r"1r'.*t"-n f"h+ d* ve*ar d4 tY Oiend;na&, tl@rt4firtrnqr"d€E:+"

kr walrrnb 'I

V = Q - y a ' b a : d r $ +i v * 5 7 r l J 5 f J f f -

,rtL-* f * = Cr*,S €X * srv,o €yl e s = - r r h g g K + c f f i e $

rrr- +??-r tr*

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Qp^ur-!;as4 #*k-ph taAil . 6-i : C'iirt LffJ ,l."

€lo'*+ic S$#Nt''A4+aa'v'v

z j : i r l o , d t - - QClrt - 7 E,1' Ert +)^ ( fir 5, * f, 'r {r< )6ij i : ) trr l i : j + 2A t)., 7-#

tll , the^u '/t4"o d.&oa

)) , Part*a ro*fo

\ nn slr.^^;n{Orrdr.","it3

7sotryi c e,(o'*;r;+ ,

ME 3+z 6r"n d a,'nPrr*oJ 1\' v'a'|i vAA Can

I-n

6,<x/f" Y Yf++

&mFffie/^Jf

.= () t zy)

l - V (T. YE oxt tr

'-YY e- I 6 x y + - : 6 y o +

e f t t t

- + 6 . n / Y 6 y v + +L

= +kn*a6yy

-:- (: I +z/ )

f 6 ,

6ut

6Li,

d z tE== z,p\ l+)))

[ry= F ez ,trz: # 6Yzf*x: + dyv

-I* *d, &Aa

6ry =_ z,F

Q+ :-7)^

t&**. s+rar.^[= i L , - i (t c-+ t)

l it \ J .\ c1(y I

K '=''',i-ll + &* )

t(rx+r7tr)= 4#r:

Gtorn t1orar,j g-rc,/\^)

L n t 4 y * f < * )

-8",{k nqirnutttrr_-

3(t->v)

Drhw il-q,u^i6I'ic

S,-, = fj

Jxn : 4," -E ,\n,/ - nn/ ,

S+re/vs tt/ruAs-r

t r 6 J^ < - T) v V = U y y - " / ,

t l ' t

(^) y s : D / * ,

) '3* : : :

?*'tY -'

C/*-n rfa,rfo-rt'C S+ratyr te"nZ nf

e . j = [ , j E3 ' r '

6rv-7 ( txx= [" , - f , €yy= t r r - { , €ra: f .**}

6rr €r/ lxl €Yu = ty = Q* = t+,x

ernd ole,vin hMl Stra,vn iJ "* "8 );ir"irll

" ! .J r . i = l , -7 , 3

)

1 wg- t q { *+\-rvL bq,trme'n dooi5$rrv{ c Stx,t5

for

t O6vVU\t fhL equzutrbvr,n h,lrLt- Q^-l- oJ:o tlu- 501,1A!- M th eta^A-'2"+U +;uny

,+n ,rr,'/*fo 6rr d*"ub'tue Sh-vsl- s-lnvn t'\ W lr'p"* h bee- an/fA'aL-

A)

&"t ( l+z/)Lyy + ) htfrw t )?qv t ) l+ 'e f^y

)z2"t t ) )v r LlrlA)[o* G" = t3o*,ce.6^,,"/,+ z

,}t{=li* hld^^t*r"-c Strl{, ( rntaun n0 l'rrr^.1- s+v*) ,

6*+6ry+fzz

o = 3 < t

n;tL p lottm-c- Sf'.a';r^' ( uoLwotv 'i J' "'(t{ d-etu'a*c-i t. ),

I'q E 3+L F*,'.' cl a merCol; Eq w r',{-iu.l'tj \ f t a

-7

3 s Yie\d C,c'rd t f,-un

vlz posnJ*e- &*,t tl,r, tlie{a r*,d,;}"s}1 ( ,:€ n&tt* tf tr:lan*ii fu-lo,,"nl;on)

ce4\ l;t e- ,@c"itd rl bJ A ftt"+frT'u n,,,L s*":'v\ {q"'t'o{

+ ( / t l ' ] ; : o , " ( ' J ; : ( r r x , u v y , ( 1 t : ' , o y r , 6 2 ^ , 6 r / ) : o

At +,tr1) foird , f r) ^ 3er"tt*l?- 1\*"(,f.,tr?L '

Motzth) , f t ,n t - o^d r i t * . .s ( r i , t =*o j , t ' r '=#\) d-o not

onJ-w fJo ttu i-r Yi/( 'j^ idz'Jiaohun ' Ad" tJ ^q'(1 i*r$ed bJ

nrylr - t f"l- sfuzrrtovhwv'; J''2 J^* l'p' ''nuh-U '

W e S,,h^-!,J* 645u-m-e- +hlt

5hlr"1"4 nat 'l^*'f b'1

f ; { \ } ) : t ' ( 1

f* uXao''ytn, t{'o-

l , ( . . 6 : t

-fh^\ rr&-attti +hsf hJe rnn,,l b" '''/uf'r' 6 UXlk' f in ++/wU

S4tvt: iitv'Nia'att fuqf,ft-''t'J "f ? th^t /'ou nnf fuuW U

Cna-&t7',,r^f-\ franAftunorA\v1'

-ilr- wia.;te;ri*L iJ isolrrrp iu, +A-p-', tl-t t^k^t "l {

0+fu:rr s/tt| t'nvaa'^'a/^t4 C-a/^ bt- "ttuf'wl in kJ?t"tuJ 4 tl4'L

pv inc, ya| sl'v"s vo'h'rL$ ' q ' 6> ' 63

Thul a,rt- thz nvan,^'(- s']i+*r vaJN4A m a s7+c''^LI

J"""*nuf rJ,g s\,sknt st'rJa th^t 'A ske*'' 5fi-e-+re-4 rt^alhu a

A cnt',,:-rli rLa-t"L +-'z'naf*n"rltrn , l.e .

6r' ) ), ,,,, 'n"/os"-t tJ' : A,if A-i t fif.

-n&,,rt\ **nnJ,- s*e-n E' =* 5,'; tJ asfw invwt;w'tJ

6 . . : + 6 ! "" t ! )

6 L

MFYII Ft,.J*t'.{,t{cL fo ttffia'5 Ca/' B

Nlo*t^r^"+,'*lt ,Stlt/vl n#*X

cI: o.A-e- #it fli,Se,ovatu+; "f +i\"l*

[ 4$^ 5xl 6xr -l

I ,T' Syv 6y* I* 4x 6+y dtt J

v l t

c \) G ; i \, J )

da 1 L r

i " e 6r , 6, ,6) &^rg

I r - i i lL t i Jdaf

th.c- t hr€r: Jo l,q-,tirrrul .+ +kL

{ rrr, - ti u nl cir,+ \l . r l

I ut;,. 6yy- ) 6'r. II or r, 61 ,ra* -i I

Pd1,1,'ot,in-8- oqwc-*irn tf iTl'\L *-S*-etiAAffi"\ DD

f Jr X €wQnq."s- ar , 1-., l-3

€,lj *. -e4 'at+rlrn

( z t1 - h l la/rQ- 5;,1"J-re^4 lhuo-nin ^fo1

Btc".*rc- 6r ,6>,6-1 one soL\ *ikv\a "f +"" dt* -e1r'Ln-tira iart mr,u1. hrvrt

& q l t i - o ' l ( 1 - o ' ) : oil - (qt o-"-t tr: ) i - (o'o" + 6'6: * rtar ) i - 6r d" 6l : o - -- - ( 3)

^f*ry %rrt a,,nd 4rl), y3o'f( l r : 6 r - t f r * q - 3 6

Lz=-S, 6r + fa trl r sr dr)

I - 3 : 6 r f r 6 3

Ttg. sfu,t irtwuo,,,^bt covq olao W c'n'';D"t'-rJ fn'*- l,t) 4 a-m n'b;"^'1

Oa>dqvfr SL,).4&e-*t ( f"l Wt'A"t\ J'L,,l't.'t Stnva:eA dn r,'ot v-rurrrrJ L Ic-omVu^>,1 \tti "^r( €7 ( t-l , *n !tJ''

l ' I ,-L,=l6vt 6yr l + lonr 6r- \ + Jo,r 5*/ I = Q*6yy+ dyrrrr+ 6.cv)

l6v u*l I e>x 61+J | 6yx 6ry1 _(syi + c*j+ 6yi )T - l f i r 6 / 6 r +L3-

I q^ 6yy fyo

Thrvep s*ren,h uen^an'b ; k)L*^a*{ tlu D } ih cr p^L s{f/o v trJ-o:r;5

) t J

lvota: io-no boolc: c]""rv12- "F'F"t"'tt tBtt')

\ in tl"t ds'f'r^n*+icrt of f'. J /

r< f. . , f , ' t -tyzx ,t l - t*

= :L(t,-. 6ii - t) "tt)

ilAe 3+L F.^^dc"rr q^d"-q- k \Ao.*8n4L

Co," 7

Eeco.^,r*'tt^* ynld co'"&t;a"r is vxyo',;rxr,,'h!-lnV fr"^il t, t ;^g^a"^rt- t'

l,vsLv\)- ( i-e /tJd"'sl+^1-" sh+44 , it iJ m u-rt cAT vw,v; e/^rf tp

+Lw lnval,i'a.,nfu I /q)tr^tTnz 5-17+ts ,!JO{.ltwJ *-C..,*rfiun f ln +un\A

o , )

Io tJ

kl 5, , St., I Lz +ht- ffincifo'l- "a-!"u 't' fuiJ

i.,c t^1 a/"t+- the- -Sal^lrrna, o ,l:

7hL etfi-Tus'a+tn

1N - _ J r t + f . l t J ; : o

wfu-u- Jr, J-, T3 o"'F€- thuon,.-amD tt o{-e-r6'tw{L s+re/4

(1 - s,) [ tr- s' ) (tr-L' ) : oP - ( S r t t + S r ) t + S , S , t S , I 3 + J r l , ) t r - 5 1 5 ' J 3 = o ( 6 )

Jx *,x/

Y'*A,t"*j l J'/, s)Jax s?

I : p-! v*t - ,^.! *+ -tr

{ SJJ

-"y""Y %(4) o^ot Et(5),

I J , - J r - t s - t s3' f iL:-F, tr- t s.s-r t srJ,)

t J3 :- sr !. 5; - i(si*

corfarig LXrH o"Yrd tX ( 6) ,

f J r

:'rj€- 3et (. 3:n*l;:i1-i*,.,+1,'+r'r.tzr,r+r / ^ 1 , r l r r L \ / + a * n c . e r r v t=::= ta,t'J+ ":l

(- :y:,,1* r* Ji-

sii,tsi,) ( -f,+st*,j:\t#'H"i'** h3*- Ho{t

\ -{ ,, J" - s slSr=-J -fr J.(J, -"-,

)!r,'r. 3er (ur.^Io- r-irrtkr :o

/

I o =[t^r+Jyy+jsr;"= r^its; + \i*\ +lJxx i"r+rJry!**r-rrrr

f. --(J xx Syy t S.1r !*o* 5** 5^') + ( 5 f- t Sui * S"i I

= +(sk+ Ivi + Sti ) '' ( tYk {- s& tJ 'o I

I : \ t ,.r, ,1.

\ ",

Y\orm b+ vt 'tov \uarrtd

fn rYY JYL II .J+, Sy 5l* I

1 \. J l

I

/I

(^' l ol/ttwL F*d"*'s^rt*-Q. ftvre*tt'n't

6

tl<t-

f-tVU- 0/rg thg S+eA rnUnUa,'nf,S 6i{) I'rn'rre- obla,,"i'A'd

Ir --36, Tr=-i(oi '61*c;,- i ,T' ') , I j=-duf(E)

J r : c , f L = | S J J j , , J 3 . : d ' e ' t t f ' ; )

yigtd c,rmdr'-fio-"<- ca'^ bL cuzi#t,rn. aa ?.lth'*rr'

{ c t , , r , , T 3 ) : o6{ t'

+ ( 6 , f " . , J r ) : o

*f*4, it w,'t M +ha'f 6 hat ntt tryfe 41""t o"'4t

ltellj-'U** (a'nd pla*2fut"^^t.* ''o geru^! "'e^\h 6nt2'r'\'et vnl'il^tu)'

Jo tfti- Tleld u'*ldt^ ( u'*Vu tzt'pAsam co'n bz un;lt<'r a't

f c L , r s ) : o

I^ thc moSf wida\ u5"4 '1ie(J con&'*-rw (l'-7\^^+afu\-

tt<r- l.zyead^n ^ I, i5 alto Jr"\y*l'

{ c f , ) - - L - t -o - l /o^ M;x t ' c ' t i }+ ' ;o " "

L* ot'urtlilnl, ,"nJ,!,(,,'t l// al ?u,ru (ii:n^dn*#xn )

tl TrV 6,V6) , 1"h-{'* 6t

k, ;l -th!, qrc^rteil+ s fu*t ttu/A '

- 63, - '1, i€' t'-fnr7^th*-r,,

r^ WA , (s,-'ii,)'- 't'k;lfrt;-s' ;'-v*;l[rs'-;;'-vt|]: Q

-) + (T,,Js) = +J -;'- tT Jr'- 3t ft;'T" + 16 W]' - 6+1 : o

1,,'lr* Irrn-^^h'$' 14 u'ull't- rn'v7t-'-^1'Y

Mdle d^'ffi a-Af h ,rt)w + f"h"pt{ Sfi+*t n'{tt- -L'*"^* J.,'o"I4

L^- pr^c,h'cL , fuctiu;$un* 6*hwp^ vrn /v13u a'nd Treacp'

c,/rfry"itu is Lp*l -(tlrrr; ts"{, utilhll tJl';c^t uyvo*J ?mqT

-f t\t_ s,w,7l* l-rr.r^ ,+ vm M\ tu.t c*i{e*'tw'. 7t5l<0a rf mui-f ,Do7''"1 wr'

3? Mohv/s attis

(',-F* &x p't' i.ivr'-r'vfo{ "&"-i' ' rp'}asvl'l tJ' fii' ftt'iz'\ C.r:,k;,;s'^n, i?-R* l,-e,^r+,"r1M& 2 )

ME 3+L F^^d xvw\a'^fu]- 4{L^h]ilt':ltl^,^' l t

V,Jo ** y{,{'J:e. tfu pnwmtttttv k t' +W 1'Yld s*az df h +tu"n|c- +?.!*'

L" +trrtvl,!- ktvf , fX^ 2 o, 6'tY:::: r^f,o* - o

6': * a,)_

Jxx :; rxx t syy =' ' -- j- 6^r,a

Jz = i(si" + ryits*l,- )= i +

k ':yie} pwt fz= * ui : k:'

6 y : J 3 k , r C!-1K- ;T1i= (V,'n nfitu )

( fu Tvusc., g'rt: (d gna J^'-*rt'a, kr : 3;

i: :r,.,f'r-ted 6 ,lrn* sh-e.o'" s'lrttA *7 G!,(6thuL si'lzrt Y'

th-e/L f z: 6- y

4t onn* r{ VNtd, 5^/=T" I' : TJ =' k'

,.- T. : k- , i.e L riJ ths- cnlrc*!- .9h-p-ort sM'$ ^rtr"'L ( V,r', Miel)A lrtittL Shuarr S*etYt iJ *Yb"eJ

C Fto',' Tru.* fl.\{ cn'r$hrsrr , l<r: T, ols' )

<-l[][-,-*.--> 6xY

W*.u^- 6v a^d ()i:r:,7, a^t th{- truug 1.,li.n-}*/ln s&v:l c rg'r^&nA/R'J

_L_3

L"1" rTxT

V"n Um \J c^:nd^*i6r4

5nl6; + o,li, = k'

L

Mil

II

;

iw

k=F: ae]ffi+k, =t ' 'o /

q--i51.d y = . k f

N^vr-lYt

5xx

l2xtn'o&t=_ &ari tlre- Cw-uL *Y*fan*]( t" tl^a- fvga6e, W(d cszt$-$7tn 611 +/tt'

\ fxx - gx/ 1tl*'r* *infr' M,'fbL

& ? : - i * '- z l - - L6,r, = T ov

ry tlt!- s^r' 7lx

6it

Tivxo'\

MF3+L Fnrla,wt*u17" q-^firy lD,4 l zTn ginQ L;l f ud On'fe,,,,r-nyr... br) EX pero, nt^e.,,t'5

Co,t*idp'- a +^lx- 9ulje"r"t',eA t, botl, PMirn f"r* tra r " d , f u I * TThr.ulrl"er^t SlwwL a,bw^(L wil( h- s"t\u^redh brtlL?tvr-LaQ st rtn Ex ,und S /rza,+' Sfu'v1 {y7.

f^urut";xg F*d T toqlrtlaL,Aht@f*y + a* oo**+,,^zfa,UnTNS u^ b, ,'hupsa+." cfy-a a," d Qt W witl, E

: :-s>yvf tMeaUnny4,w ";hZ{,,rr,-truns { 6<x, A* F WttwwS ?* ia/,%u6 ,,^l lp m^f tts j/htd Swr,(-r-e- o^rl u*Wv t-th flr$-phb! drh .

* Vrn ltth'sp,s ortfu'l"n Tp17-,,*- b "g+.n t**rr wr+l+ %PUit^QlfE +he^ ftWraia Ou i--'l

Yo

Trescd law Misss law

o CoPP',X Aluminiumo Mrld Srcel

fFro. {. Experirnontal recults of Taylor and Qrriruney from combinod toraion sndtension tsets, each motsl being work.hardonrxl rge tho samo ststn for all 0eet8.Tho Miees law is ot*3rr : yt, whilo tho I'rcpce law ig orl4tr - Yr, whoro

o : ten-silo atreaa, r -- ahear atresS, .ll' ,: teneile yiold s11,r988.

Fto;,P. H;U, Tl',n-lttodttnc*hUtLr/-%{ltr,,rn*2xN (r;, Prt >7"/75 0P.2L

To/n, a^ol Qu;"Y'Ypld(, Tr^"a kj5oc, A L3o,)23 ( l?3t )

.5

,4

.3

-2

' l

. 9'8,7,4'3,z. l r.0

ME 3+L r - t r A -iwn d o/r'\ti w*tNl fi t t a*t}rrt,S

t

C"" r j

tq Ca!4,,h-{ rl\rq- (Sirer4{-Jnra^4^) &e[ot#in4 in thr Pl@

5" ,^sdb , h.att wL cl-,nz,,.&.,.u ^ y$^lr\ pl"*tt. -aon^[ cno s1qin havLda,.'v1

TL*^ ;n tb pta,r+Z reyta {Uo;t')): J,-t: o, L rv^"'^ c.anrla4nr-

i .e. ! r : t , j , : o

dlau.twe4 tlrD "Ine'd naf rnz'a,n tha* a// s+ruv ctYr'p'*ta slalz cni"'"'*

I, - kt ^' ,1^'e[J snj"u-' ln tb s+t'+t tface-

7hz s+w ^rf"ffi ^ wU d+Lin +A* TVld tW

Rxro,!,(- 7, : 4$h + sy', -rS'-+* ) + t\in 1*i+l*i )

ir= f^r 5x" t l,1y iyy t Szti* t rEzls tr 5+"J*, t.l 5x/ty : sg'sr': o-nM, k a, c.a,t*tral-,t on+Az stre* mte Sj

NhnL" $w sfre/ SW M +h!-

p{a*frv Strar'o *; , So th^t

g^f /e'rna; f of(n+iv^J

W'td 5v'n{aw,

ll;a^.y: ttf *

t" ,.tiWyt s-1

n^rfu?r {- w')it xndtryo

t, tfl E;iJ,) = tU €r1

tl,l*s-a ' *r-4 |-,

: + zru e(

Tt +r\k colq&

Il

!,

Lt becan'*fu,twn''ngJ- r r l

Lnsteaa ,

Q-= Haro^ )

h dPlu>rrv;fie- 8,!i'

+iy/\!- fu 5q2"Jtil?5 b' f,.d o'^/r th-s- c'uLnL^+ -ry h

( *lfr, -lnerJ ;nYnq sto"r'6 ) '

,] J

-?L )r L',.t*mp> dLwep'-k, t"€' ' ' f Cannot bg+n!6 .J is ,hdahrY' "-l- . '

thL UeL!'v\f Sl^rai$ S-tatu ( ttt )'

rnwy(be-cl-p ' tw,n-. , ,^ ld,JACCUtrtwl"AnftSnLJltTlcz|-r l tpnf i5.€:;

I^ a,Jhm, i? ln'"t fuf' f-^ai' tj'vt pta'>R' ry PIod^'so

*f g;bl.!- val^^rN- cj\tur\&L ' € iPiL:o, i:: - o

"tzufw'1o-n-c- pla,+ht. Str-alt )'til-o- -lc,rg- ^-o pl.a":lr''L }k--'u)'

h"*a

" ? L-J

piL$rt

UrS. P L.ij

ME3+L F*^d*^{^t^[ Eqwahbn'xr Gu t+A,ts**yhrrt

*A*+h;t, ,,$L 2y nf := -i)' )

NoR. bd\" tlre ds'aatlni<' ela'+t.'z S+Ylin

r"ftffffi\ the- de4^u,'+lYui:c- s+e,etz/ery : ,u(dff" e,f') =: .s,i + ls,r

T:ft,;,/

ov'ctittL'k 5+aud oA Ca*z'?'^uw ). ) , - :

' , ; t ' - ) , U ; - r -

' '

W =Jxx (xx *Syl 4y t Sz+ d** + ,. i Sf*ey+ + a J2* €*x +)ixy*,

:o f* Fr"le^H"U Plw*z r'tr'ne'xa"l

^'-Sii - zT -'l''- :- zT *) " )

Wftlrrr. ,'yVt4 co"1d4't56Y\ it 't'tzaclsd ) i{: Wfi il!. ch,e,,hw,f ri,

,rrr.{"-q- }" iJ a. pos;Nt- scoJ*{ {-,i",/tF bq- rL{",tg:u'ra,i*rd

, a 0 .

i,.,u ?,1 : J,'-- ' J )

1 F= A J r ' iJ2y e;'

?Ad, +lw plt/>'Fv slra,)*4-';f*lL

@ t )( €,51 t( J,-t ')

, '6l = S,, €rJ'

. e I- - - I i i fe t i +,J ' J

2/"

kecaft

W : 5 ; ( S , , j + ) , r l )

- T - c - C ^ ,Jz - - t ; j ' ) '

2 r ?2 / N) / 1

r',-l!. tlo"; t, oklo-*'"n fi; (-**-+*.**

s,.f'- {r. s.; , J',)-- ,lt i( - 'y(e",-a!,!)" ' l z { J )

rr^t \tnstan s*ut +)ra,h *x:, f$' ; s''t' >*;rf, rxwa - - 3 K 8 , o = 3 k 6 / t + L r - - f u L , ,

T dl,l*an;nw d/-2- f-"tt-t X , ^/Q' n'a2/ to inttp duco'

<- Y*tt, ':f Na?k l-ovti-- ax)ocr^fteA ':' shy+ eV

. Dattf ''of

^l';'l- 1ot% It stnzt' elashL %u^3y

i'i,\) ' p^fi ,f *AnrJ' C^tt;y",ted ^4 t"k''f

(w,r.l,. J'r','!- a''vnaz-tqd wftLwLww c'Wry+ A

plt3*7, Furdo''rraatfxf h,,7' vr,x*ilsrui G*t t5

Sw^ n',*ut. Llta 1MAS , t*nd*,,, a. ryec,.t';ud clr{rQ ) Jkah, rch i

t/r-a- plw'u*L teJ ri,vrtt- !

r* d a*u"r'nt 9ru4<l Oi,' , b

r -l-.. L7 : J- 9..- *--> (J' :=:= J K €\-

3 (-f (

i'n

GtW,r,_ f', -' q J

i , 7

'''J -.--=-*

5 to Su-rtttran3 "f

Earttn R"ryq =

)ied c*trrd^f-s '

Ptq*'c Hft*.

J ^ r | / ' , / ) f-f;nd 6;i ( ifn^Aifte- ^ rn-al4-L f-ai,n ntJ' 'J p rrien S----__,\

; . )g , ' ) = S )+66 !

6,j

\ ' l n , ' t , /-"' J) : rJ Vj-yf tla'.

f f : 3 K tSii : zf e;i

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ME333B Elasticity and Inelasticity – Wei Cai – Stanford University – Win 2017

Handout 15. Tension and Shear

February 12, 2017

We now work out a simple example problem to show how to use the yield condition andflow rule to predict plastic strain along a deformation path. This example also illustrates thehistory-dependent nature of plastic deformation. This example is extracted from Ref. [1].

1 Problem Statement

Consider a block of material to which both tensile and shear strain can be applied. This canbe achieved in a tube that is subjected to tension and torsion. We shall assume that �

xx

and�

xy

are the only two non-zero stress components, as shown in the diagram below. Noticethat the elastic strain will have non-zero yy and zz components due to Poisson’s e↵ect.

For simplicity, we shall assume that there is no work hardening. We shall use the vonMises yield criterion and the associated flow rule. We shall also assume that the materialis incompressible, i.e. ⌫ = 0.5. We shall see that the incompressibility assumption usuallyleads to great simplifications in the analysis of plasticity problems. We shall assume the vonMises yield condition (J

2

= k

2) and the associated flow rule. For simplicity, the materialsassumed to be non-hardening, i.e. k stays constant in the plastic regime.

Due to the incompressibility assumption, the elastic strain "

el

ij

also conserves the volume(the same as plastic strain), so that

"

el

ij

= e

el

ij

(deviatoric) (1)

Given that the plastic strain always conserves the volume,

"

pl

ij

= e

pl

ij

(deviatoric) (2)

1

we have,"

ij

= "

el

ij

+ "

pl

ij

= e

ij

= e

el

ij

+ e

pl

ij

(3)

We shall consider two di↵erent strain paths as shown below:

I. OA-AD: the block is first loaded in tension to exactly the yield condition and thenloaded in shear

II. OB-BD: the block is first loaded in shear to exactly the yield condition and then loadedin tension

Our goal is to obtain the stress as a function of strain along these two strain paths. As aresult of the incompressible assumption, the Young’s modulus is E = 2(1+ ⌫)µ = 3µ, whereµ is the shear modulus and ⌫ is Poisson’s ratio.

2 Strain Path I: Tension-Shear

We first consider path OA. By construction, all strains are elastic along path OA. Therefore,

�

xx

= E "

xx

�

yy

= �

zz

= �

xy

= �

yz

= �

zx

= 0

"

yy

= "

zz

= �⌫�

xx

= �1

2�

xx

(4)

Notice that at point A, we have�

xx

= �

Y

=p3 k

"

xx

=

p3 k

E

(5)

We now consider path AD. It is reasonable to assume that the material stays in the plasticflow regime along path AD. This means that the stress stays on the yield surface and theplastic strain rate is non-zero. Because the material is non-hardening, we have

J

2

= k

2

, J

2

= 0 (6)

2

The plastic strain rate is

"

pl

ij

=W

2k2

s

ij

(7)

where W = s

ij

e

ij

is the rate of work done associated with shape change. Notice that inthis example, W equals the total rate of work done W

tot

because the volume stays constant.Therefore,

W = W

tot

= �

xy

· 2 "xy

(8)

Notice that�

yy

= �

zz

= 0 , � =�

xx

3

s

xx

=2

3�

xx

, s

yy

= �1

3�

xx

, s

zz

= �1

3�

xx

J

2

=1

2(s2

xx

+ s

2

yy

+ s

2

zz

) + (s2yz

+ s

2

xz

+ s

2

xy

)

=1

3�

2

xx

+ �

2

xy

= k

2

(9)

The total strain rate along path AD consists of contributions from the elastic and plasticstrain rates.

"

xy

= "

el

xy

+ "

pl

xy

=�

xy

2µ+

W

2k2

�

xy

(10)

Plugging in the expression for W , we obtain

"

xy

=�

xy

2µ+

"

xy

k

2

�

2

xy

✓1�

�

2

xy

k

2

◆"

xy

=�

xy

2µ

(11)

As long as 1� (�xy

/k)2 is non-zero, we have

2µ"

xy

k

=�

xy

/k

1� (�xy

/k)2(12)

It can be verified that the solution to this equation is

2µ"

xy

(t)

k

= atanh

✓�

xy

(t)

k

◆(13)

where atanh is the inverse of the tanh function. (Note that datanh(x)/dx = 1/(1� x

2).) Inother words,

�

xy

(t)

k

= tanh

✓2µ

"

xy

(t)

k

◆(14)

Given the yield condition1

3�

2

xx

+ �

2

xy

= k

2 (15)

3

we can solve for �xx

along path AD now that �xy

is obtained.

�

xx

k

=p3 ·

r1�

⇣�

xy

k

⌘2

�

xx

k

=

p3

cosh (2µ "

xy

(t)/k)

(16)

The stresses as functions of strain along path AD are plotted in the following diagram.

We can see that at point A, �xx

=p3k and �

xy

= 0. With increasing shear strain "

xy

, thenormal stress �

xx

decreases while the shear stress �xy

increases.

At point D, �xx

= 1.12 k and �

xy

= 0.76 k.

If the shear strain were to continue to increase beyond point D, eventually in the limit of"

xy

! 1, we have �

xx

! 0 and �

xy

! k (which satisfies the yield condition in pure shear).

3 Strain Path II: Shear-Tension

We first consider path OB. By construction, all strains are elastic along path OB. Therefore,

�

xy

= 2µ "

xy

�

xx

= �

yy

= �

zz

= �

yz

= �

zx

= 0

"

xx

= "

yy

= "

zz

= "

yz

= "

zx

= 0

(17)

Notice that at point B, we have�

xy

= k

"

xy

=k

2µ

(18)

We now consider path BD. Again we shall assume that the material stays in the plastic flowregime along path BD. The plastic strain rate is

"

pl

ij

=W

2k2

s

ij

(19)

4

where W = s

ij

e

ij

is the rate of work done associated with shape change. In particular,

"

pl

xx

=W

2k2

s

xx

=W

2k2

2

3�

xx

=W

3k2

�

xx

(20)

Because the volume stays constant,

W = W

tot

= �

xx

"

xx

(21)

The total strain rate along path BD consists of contributions from the elastic and plasticstrain rates.

"

xx

= "

el

xx

+ "

pl

xx

=�

xx

E

+W

3k2

�

xx

(22)

Plugging in the expression for W , we obtain

"

xx

=�

xx

E

+"

xx

3k2

�

2

xx

✓1� �

2

xx

3k2

◆"

xx

=�

xx

E

(23)

As long as 1� (�xx

/

p3k)2 is non-zero, we have

E

"

xxp3k

=�

xx

/

p3k

1� (�xy

/

p3k)2

(24)

It can be verified that the solution to this equation is

E

"

xx

(t)p3k

= atanh

✓�

xx

(t)p3k

◆(25)

In other words,�

xx

(t)p3k

= tanh

✓E

"

xx

(t)p3k

◆(26)

Given the yield condition1

3�

2

xx

+ �

2

xy

= k

2 (27)

we can solve for �xy

along path BD now that �xx

is obtained.

�

xy

k

=

s

1�✓

�

xxp3k

◆2

�

xy

k

=1

cosh�E "

xx

(t)/p3k

� =1

cosh�p

3µ "

xx

(t)/k�

(28)

The stresses as functions of strain along path BD are plotted in the following diagram.

5

We can see that at point B, �xx

= 0 and �

xy

= k. With increasing shear strain "

xx

, thenormal stress �

xx

increases while the shear stress �xy

decreases.

At point D, �xx

= 1.31 k and �

xy

= 0.64 k.

Notice that the stress state at point D is di↵erent from that obtained along the loading pathOA-AD. This illustrates the history dependence of plastic deformation. If the normal strainwere to continue to increase beyond point D, eventually in the limit of "

xx

! 1, we have�

xx

!p3k and �

xy

! 0 (which satisfies the yield condition in uniaxial tension).

Exercise: plot plastic strain "

pl

xx

and "

pl

xy

as functions of the total strain.

References

1. Dr. Paul Paslay’s lecture video, “Introduction to Theoretical and Applied Plasticity”.To receive DVD copies of this video, send email to info@blace-energy.com

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ME333B Elasticity and Inelasticity – Wei Cai – Stanford University – Win 2017

Handout 18. Hardening Laws

February 17, 2017

So far we have considered elastic-perfect plastic models in which the yield surface remainunchanged during plastic deformation, i.e. no strain hardening. We now consider variousways to include strain hardening into the constitutive equation.

1 Plastic Instability in Tension

As an introduction, we will first show that a perfect plastic material (no strain hardening)is actually unstable when loaded in uniaxial tension. It will immediately experience neckingafter yielding. In fact, the strain hardening rate (d�/d") need to be su�ciently large forstability against necking. Most ductile material loaded in tension eventually fail by necking,when the strain hardening rate drops below the critical value. The following figure show thestress-strain curves of a elastic-perfectly plastic model (dashed line) and a strain-hardeningmodel (solid line).

We now derive the stability criterion against necking during tensile deformation. Let A0

and l

0

be the cross sectional area and length of the rod prior to deformation (i.e. in thereference state), respectively. Let A and l be the corresponding values in the deformed state(assuming necking has not occured yet). For simplicity, we shall neglect elastic strain. Thismay be justified because the plastic strain can often become much larger than the elasticstrain for a ductile material.

1

Because plastic strain conserves volume, we have

A · l = A

0

· l0

d(A · l) = Adl + l dA = 0

dA

A

= �dl

l

(1)

This equation expresses the relation between the reduction of the cross section area and theincrease of the length of the rod. Recall that the engineering strain is defined as

" ⌘ l � l

0

l

(2)

Therefore,l = l

0

(1 + ")

d" =dl

l

0

dl = l

0

d"

dA

A

= �dl

l

= � l

0

d"

l

0

(1 + ")=

d"

1 + "

(3)

This equation expresses the relation between the reduction of the cross section area and theengineering strain of the rod.

We now establish the requirement that for a rod to be stable against necking, the requiredtensile force F to plastic deform the rod need to increase with an increase with the lengthl, i.e. dF/dl > 0. This requirement can be understood by considering the following process.Consider a rod subjected to tensile force F , which brings the rod to the yield condition. Nowimagine that, by some random perturbation, a small section of the rod undergoes plasticflow before other sections and experiences a small increase in its length. (For example, thiscan happen if a small section of the rod has a slightly lower yield stress so it yields slightlyearlier than other sections). If by increasing its length the force required to further deformthis section actually decreases (i.e. becomes lower than F ), then this section of the rod willdeform further (becoming even longer), and the force it can resist will become even smaller.This creates a “run-away” process, i.e. a positive feedback loop, with the net result beingall the plastic deformation is localized in the small section that happen to yield first.

If, on the other hand, dF/dl > 0, then the small section that happens to yield firstwill require a higher force to continue the plastic deformation. The result is that plasticdeformation will stop momentarily at this section, and other sections will take their turn todeform plastically. The net result is that plastic deformation will occur uniformly along theentire length of the rod.

The axial force F is simply the product of the cross section area A and the axial stress�, F = A �. Therefore,

dF = Ad� + � dA

= A

✓d� +

dA

A

�

◆= A

✓d� � dl

l

�

◆

= A

✓d� � d"

1 + "

�

◆= Ad"

✓d�

d"

� �

1 + "

◆(4)

2

Because dF/dl > 0, dF/d" > 0, we must have

d�

d"

� �

1 + "

> 0 (5)

d�

d"

>

�

1 + "

(6)

This is called the Considere condition. It means that the strain hardening rate must exceed�/(1 + ") to avoid necking instability. This condition has the following graphical represen-tation.

Consider a stress-strain curve, �("), which has a concave shape as shown below. We candraw a series of straight lines all passing through the point " = �1, � = 0 and intersectingthe stress-strain curve. One of the straight line is tangent to the �(") curve. At the tangentpoint, d�/d" = �/(1 + "), which is the critical condition, i.e. the unset of necking. To theleft of the tangent point, d�/d" > �/(1 + "), the rod is stable against necking. To the rightof the tangent point, d�/d" < �/(1 + "), the rod is unstable against necking.

As an example, we now give a rough estimate of how much strain hardening is needed ina typical metal for it to be stable against necking. Suppose the metal has an yield stress of�

Y

= 100 MPa and a Young’s modulus of E = 200 GPa. Therefore, the strain at which yieldoccurs is approximately "

Y

⇡ �

Y

/E = 0.05%. Because the stress-strain curve bends abruptlynear the yield point, the tangent point between the straight line and the stress-strain curveis close to " = "

Y

. Therefore, the critical strain hardening rate is approximately

⇥ ⌘ d�

d"

⇡ �

Y

1 + "

Y

⇡ �

Y

= 100MPa (7)

This is a fairly low strain hardening rate. For example, with this hardening rate, the flowstress at strain " = 10% will be about 1.1 �

Y

, as shown in the diagram below. This explainswhy a material may satisfy the Considere criterion for stability against necking, and yet canstill be well approximated by the elastic-perfect plastic model.

3

2 Isotropic Hardening

With isotropic hardening, we express the yield condition as

f(�ij

, q) = f(�ij

)� �(q) = 0 (8)

where �(q) is some hardening function, and q is a measure of work hardening, and f(�ij

) isour yield criterion without hardening.

For example, in J

2

plasticity, f(�ij

) = J

2

� k

2, so that

f(�ij

, q) = J

2

� [k2 + �(q)] = 0 (9)

Therefore, the e↵ect of �(q) is to modify the yield parameter k2 isotropically, i.e. the entireyield surface expands uniformly in all directions. As shown in the diagram below, plasticflow at one point on the yield surface results in expansion of the yield surface everywhere.

The following are a few common examples of q.

q =

Z�

ij

d"

pl

ij

(10)

is a measure of the plastic work.

q =

Z r2

3d"

pl

ij

d"

pl

ij

=

Z r2

3||"pl|| dt (11)

4

is a measure of the equivalent plastic strain.As an example, consider J

2

plasticity with a linear hardening law where �(q) = � q andq =

R�

ij

d"

pl

ij

. We consider how the stress-strain curve would look like if this material issubjected to unaxial tension. The stress state under unaxial tension (along x-axis) is

�

xx

> 0 , �

yy

= �

zz

= 0 , � =1

3�

xx

(12)

Therefore,

s

xx

=2

3�

xx

, s

yy

= s

zz

= �1

3�

xx

J

2

=1

2s

ij

s

ij

=1

3�

2

xx

(13)

From the plastic flow rule, d"plij

= (�/2µ)sij

dt, we have

d"

pl

xx

=�

2µ

2

3�

xx

dt , d"

pl

yy

= d"

pl

zz

= � �

2µ

1

3�

xx

dt (14)

The accumulated plastic work can be expressed as

q =

Z�

ij

d"

pl

ij

=

Z�

xx

�

2µ

2

3�

xx

dt =

Z�

3µ�

2

xx

dt (15)

From the hardening law, J2

� [k2 + �(q)] = 0, we have

1

3�

2

xx

= k

2 + �q

2

3�

xx

d�

xx

= � dq = �

�

3µ�

2

xx

dt

d�

xx

= �

�

2µ�

xx

dt

(16)

Combining this expression with Eq. (14), we obtain

d�

xx

d"

pl

xx

=3�

2(17)

To determine the slope of the stress-strain curve, we need to consider the total plastic strain,"

xx

= "

el

xx

+ "

pl

xx

. Because "

el

xx

= �

xx

/E, we have

d"

xx

d�

xx

=d"

el

xx

d�

xx

+d"

pl

xx

d�

xx

=1

E

+2

3�(18)

Therefore the overall hardening rate in uniaxial tension is

⇥ ⌘ d�

xx

d"

xx

=

✓1

E

+2

3�

◆�1

(19)

5

Plastic deformation during tensile loading increases the flow stress top3k2 + �q, which is

higher than the initial yield stress, �Y

=p3k. When the tensile load on the sample is

removed and the sample is loaded again in tension, the new yield stress isp

3k2 + �q. Thesame yield stress will be measured if the sample is loaded in compression, as shown in thediagram below. Because the yield stress remain unchanged under reversed loading, there isno Bauschinger e↵ect for a material with isotropic hardening.

3 Kinematic Hardening

Instead of allowing the yield surface to expand isotropically, in kinematic hardening weassume the yield surface to translate in the stress space by plastic deformation while keepingits size unchanged. The yield condition can be written as,

f(�ij

, q

ij

) = f(�ij

� q

ij

) = 0 (20)

where q

ij

is the back stress tensor that describes the translation of the yield stress in thestress space, as shown below.

As an example, consider J

2

plasticity with a kinematic hardening law where q

ij

= c "

pl

ij

where c is a constant. The e↵ect of c can be understood by considering stress-strain curvewhen this material is subjected to unaxial tension. Again, the plastic flow rule gives

d"

pl

xx

=�

2µ

2

3�

xx

dt , d"

pl

yy

= d"

pl

zz

= � �

2µ

1

3�

xx

dt (21)

6

Therefore, the back stress tensor can be expressed as

q

xx

= c "

pl

xx

, q

yy

= �1

2c "

pl

xx

, q

zz

= �1

2c "

pl

xx

, (22)

The new yield condition is

f(�ij

� q

ij

) =1

2

"✓2

3�

xx

� c "

pl

xx

◆2

+ 2

✓�1

3�

xx

+1

2c "

pl

xx

◆2

#� k

2 = 0 (23)

1

3

✓�

xx

� 3

2c "

pl

xx

◆2

= k

2 (24)

�

xx

=p3k +

3

2c "

pl

xx

(25)

Therefore,d�

xx

d"

pl

xx

=3c

2(26)

and the overall hardening rate in uniaxial tension is

⇥ ⌘ d�

xx

d"

xx

=

✓1

E

+2

3c

◆�1

(27)

Therefore, c in the kinematic hardening law plays the same role as � in the isotropic hardeninglaw under uniaxial tension. However, if a material satisfying the kinematic hardening law isfirst loaded in tension and then in compression, the yield stress in compression will be lowerthan the original yield stress in tension. In other words, forward loading reduces the yieldstress in reverse loading, as shown below. This phenomenon is called the Bauschinger e↵ect.

It is possible for a material to exhibit both isotropic and kinematic hardening. In this case,the two hardening laws can be combined to give the following form of yield condition,

f(�ij

, q, q

ij

) = f(�ij

� q

ij

)� �(q) = 0 (28)

7

4 Loading and Unloading Conditions

When the stress state satisfies the yield condition, i.e. f(�ij

, q, q

ij

) = 0, it is important todistinguish the several di↵erent scenarios the material may be in.

• f < 0, � = 0. The stress is detaching from the yield surface. This is called elasticunloading. The plastic strain rate is zero and the total strain rate equals the elasticstrain rate.

• f = 0, � = 0. The stress remains on the yield surface and the plastic strain rate iszero. This is called neutral loading.

• f = 0, � > 0. The stress remains on the yield surface and the plastic strain rate isnon-zero. Notice that in this case we must have � > 0, i.e. the plastic strain rate flowsin the same direction (i.e. not in the opposite direction) as the deviatoic stress. Thisensures that the rate of plastic work is always positive.

For the plasticity theory to not violate the thermodynamic laws, f and � must satisfy one ofthe three conditions given above. This constraint is summarized by the Karush-Kuhn-Tuckerconditions

� � 0 , f 0 , � f = 0 (29)

and the consistency condition� f = 0 if f = 0 (30)

5 Associative Flow Rule

So far we have only considered the Prandtl-Reuss flow rule,

"

pl

ij

=�

2µs

ij

(31)

We have said that the Prandtl-Reuss flow rule is an associative flow rule to the von Mises(J

2

) yield condition. We now explain what associative flow rule means.Recall that the yield condition can be written in the general form of f(�

ij

) = 0. Let usknow introduce another function of the stress, g(�

ij

), called the plastic potential, and definea general flow rule as

"

pl

ij

= h

@g

@�

ij

(32)

where h is an arbitrary parameter (similar to �) to be determined. If g(�ij

) = f(�ij

), i.e.when the yield condition is used as the plastic potential, then the flow rule is said to beassociative (to the yield condition).

As an example, consider the von Mises yield condition, where

f(�ij

) = J

2

� k

2 =1

2s

ij

s

ij

(33)

8

The associative flow rule would be

"

pl

ij

= h

@g

@�

ij

= h

@f

@�

ij

= h s

ij

(34)

This is equivalent to the Prandtl-Reuss flow rule if h = �/2µ.

Q: What is the associative flow rule for the Tresca yield condition?

Since @f/@�

ij

points in the direction orthogonal to the yield surface, the plastic strainincrement will also be. This property is called the normality rule of plastic flow.

References

1. L. M. Kachanov, Fundamentals of the Theory of Plasticity, Dover (2004).

9

ME333B Elasticity and Inelasticity – Wei Cai – Stanford University – Win 2017

Handout 20. Crystal Plasticity

February 23, 2017

We now take a look at the fundamental physical mechanisms that give rise to plasticdeformation in metals and alloys. We would like to see how continuum plasticity models(such as von Mises yield condition and Prandtl-Ruess flow rule) at the macroscopic scalemay appear as a consequence of these microscopic processes.

1 Polycrystal

A bulk metal is a polycrystal consisting of many single crystal grains (roughly on the order of1 to 10 µm in size) separated by grain boundaries, as shown in the diagram below. In general,each single crystal grain is elastically and plastically anisotropic. But the aggregate of manygrains usually appear to behave isotropically at a length scale much larger than individualgrain sizes. If we want to understand polycrystal plasticity, a reasonable approach is to firstunderstand single crystal plasticity, and then perform the process of “homogenization”.

For simplicity we may assume that the grain boundaries do not play an active role inplastic deformation, other than providing a constraint on the compatibility between plasticstrain of neighboring grains (G. I. Taylor 1938). This assumption is confirmed by experiments

1

of polycrystals under ambient conditions. At very high temperatures, grain boundary slidingand grain boundary di↵usion contributes significantly to plastic deformation (in a processcalled creep).

2 Slip Systems in Single Crystals

A large number of engineering materials (metals, semiconductors) are crystals with a cubiccrystal structure, with either a face-centered cubic (FCC) lattice or a body-centered cubic(BCC) lattice.

If we put one (and the same type) atom on each FCC or BCC lattice site, we get an FCCor BCC crystal. For example, Cu, Al, Au are FCC crystals, and Fe (at room temperature),W, Mo are BCC crystals. We can also put multiple atoms at each lattice site to get di↵erentcrystal structures. For example, the diamond cubic structure has a two-atoms basis (madeof atoms of the same species) at each site of the FCC lattice. Here we will focus on simplecrystal structures (FCC and BCC) with only one (and the same type) atom at each latticesite.

When a single crystal is loaded in tension, plastic deformation is known to occur throughslip on crystallographic planes along specific directions, as shown below. Optical microscopycan be used to reveal slip traces on the surfaces of plastically deformed crystals.

2

The slip directions are the shortest lattice repeat vectors. The slip planes are generally theplanes with the highest inter-planar spacing (hence largest in-plane atomic density) thatcontain the slip vectors.

The combination of slip planes and slip vectors is called a slip system. In FCC metals,the dominant slip system is on {111} planes with 1

2h110i slip directions; it can be written as{111}h110i for short, as shown below. In BCC metals, slip is often observed along 1

2h111idirections and on {110} planes, although slip on other planes (such as {112}) and {123} areobserved as well.

Q: How are slip systems are there in FCC and BCC crystals, respectively?

3 Schmid Factor

Consider a unaxial tensile test of a single crystal as shown below.

3

Let T be the unit vector along the tensile axis, n be the unit vector of the slip plane normal,b be a unit vector along the slip direction, � be the magnitude of the tensile stress. Let �be the angle between T and n, i.e. cos� = (T · n), and � be the angle between T and b, i.e.cos� = (T · b). Let ⌧ be the resolved shear stress on plane n and along the b direction. Let� be the shear strain on plane n along b. Then instead of plotting the normal stress-straincurve, �("), we can plot the shear stress-strain curve, ⌧(�), as shown above.

It can be shown that the resolved shear stress ⌧ is related to the normal stress � throughthe following relation,

⌧ = S · � (1)

whereS = cos� cos� (2)

is called the Schmid factor. It is obvious that 0 < S < 1. In fact, it can be shown that themaximum possible value for the Schmid factor is S = 0.5. The Schmid factor S depends onthe relative orientation of the loading axis with respect to the slip system.

Intuitively, we expect slip to occur when the resolved shear stress ⌧ exceeds a thresholdvalue (called the critical resolved shear stress, CRSS). Then we would expect the yield stress�Y of a single crystal to be dependent on the orientation of the tensile axis (but should notdepend on whether loading is in tension or in compression). In fact, we would expect theproduct of the Schmid factor of the dominant slip system (the one with the largest Schmidfactor) and the yield stress to be a constant for single crystals of di↵erent orientations. Thisis indeed observed in FCC and hexagonal close packed (HCP) metals, and is called theSchmid law. BCC metals, on the other hand, do show deviations from the Schmid law, aswell as asymmetry between tension and compression.

4 Slip by Dislocation Movement

4

It is now known that slip between crystallographic planes do not occur uniformly across theentire slip plane. Doing so would require too much energy as all bonds across the slip planewould need to break at the same time. Instead, slip always occurs first in a small area on theslip plane; the slipped area expands under applied stress and can eventually spread over theentire slip plane. When that is accomplished, the half-crystal on one side of the slip planewill have slipped by one lattice vector relative to the other half. The boundary between theslipped and unslipped area on the slip plane is a line defect called a dislocation.

When the dislocation line is perpendicular to slip direction, it is called an edge dislocation.When the dislocation line is parallel (or anti-parallel) to the slip direction, it is called a screwdislocation. In general the local dislocation line direction may be neither perpendicular norparallel to the slip direction; in this case, it is called a mixed dislocation. The figure aboveshows the motion of an edge dislocation from one side of the slip plane to the other side,eventually causing the entire upper half of the crystal to slip over the lower half by onelattice vector.

5 Elastic Field around an Edge Dislocation

The following figure shows an edge dislocation whose slip plane is the x-z plane, i.e. the slipplane normal is the y-axis.

x

y

z

r x,yb

θ

The elastic field of such a straight edge dislocation in an infinite medium is described by thefollowing Airy stress function.

� = � µb

2⇡(1� ⌫)r ln r sin ✓ (3)

The stresses in cylindrical coordinates are,

�

rr

= �

✓✓

= � µb

2⇡(1� ⌫)

sin ✓

r

�

r✓

=µb

2⇡(1� ⌫)

cos ✓

r

(4)

5

and the stresses in Cartesian coordinates are,

�

xx

=@

2�

@y

2= � µb

2⇡(1� ⌫)

y(3x2 + y

2)

(x2 + y

2)2

�

yy

=@

2�

@x

2=

µb

2⇡(1� ⌫)

y(x2 � y

2)

(x2 + y

2)2

�

xy

= � @

2�

@x@y

=µb

2⇡(1� ⌫)

x(x2 � y

2)

(x2 + y

2)2

(5)

Because this is a plane-strain problem, the normal stress in the out-of-plane direction is,

�

zz

= ⌫(�xx

+ �

yy

) = � µb ⌫

⇡(1� ⌫)

y

x

2 + y

2(6)

The displacements in cylindrical coordinates are,

u

r

=b

2⇡✓ cos ✓ � b

4⇡(1� ⌫)sin ✓

(1� 2⌫) ln r � 1

2

�

u

✓

= � b

2⇡✓ sin ✓ � b

4⇡(1� ⌫)cos ✓

(1� 2⌫) ln r +

1

2

� (7)

and the displacements in Cartesian coordiantes are,

u

x

=b

2⇡

✓ +

1

4(1� ⌫)sin 2✓

�

u

y

= � b

2⇡

1� 2⌫

2(1� ⌫)ln r +

1

4(1� ⌫)cos 2✓

� (8)

6 Dislocation Dynamics and Crystal Plasticity

Dislocations are line objects in 3D. They can be observed in transmission electron microscope(TEM). The following figure is a TEM micrograph showing dislocation line tangles in a Fe-35% Ni-20% Cr alloy [1].

6

Dislocation Dynamics (DD) is a method that simulates the motion of the dislocation linesdue to the applied stress as well as interaction forces between dislocations, and predictsthe stress-strain curves of single crystals in the plastic regime. The following figure showsDD predicted dislocation structure and stress-strain curves of single crystal Cu deformedalong the [001] direction. The simulation cell is a cube of size 15 µm, subjected to periodicboundary conditions in all three directions. Notice that a strain hardening behavior ispredicted; this is due to the increase of dislocation density with plastic strain making itmore di�cult for the dislocations to move through the crystal.

0 0.2 0.4 0.6 0.8 1 1.20

2

4

6

8

10

12

14

Shear strain (%)

She

ar

stre

ss (

MP

a)

[001] Direction

ε = 100 s−1.

ε = 1,000 s−1.

References

1. W. Cai and William D. Nix, Imperfections in Crystalline Solids, Cambridge UniversityPress (2016).

2. R. B. Sills, W. Kuykendall, A. Aghaei and W. Cai, Fundamentals of Dislocation Dy-namics Simulations, in C. R. Weinberger and G. J. Tucker (Eds.), Multiscale MaterialsModeling for Nanomechanics, (Springer, 2016).

3. ParaDiS. Parallel Dislocation Simulator. http://paradis.stanford.edu

7

ME333B Elasticity and Inelasticity – Wei Cai – Stanford University – Win 2017

Handout 22. Slit-like Crack

February 28, 2017

To start our discussion of fracture mechanics, we now consider the stress, displacementand enthalpy of a slit-like crack in an infinite plate in plane-strain condition. In this lecture,we assume the plate is a purely elastic material. From the discussions on wedges and notches,we expect the stress field to be singular at the crack tips as � ⇠ 1/

pr, where r is the distance

to the crack tip. The amplitude of the singular stress field is called the stress intensity factorK

I

, which was left as an unknown parameter in the discussions on wedges and notches. Wenow want to find the magnitude of K

I

in terms of the applied load S on the plate. We alsowant to find the crack opening displacement, d(x), i.e. how much of a gap is there betweenthe two crack surfaces. We will show that the crack opening displacement allows us to obtainan explicit expression of the enthalpy of the crack.

1 Problem Formulation

As shown in the diagram above, we consider a crack that lies along the x-axis, from x = �ato x = a. The plate is subjected to tensile load in the y direction, i.e. �

yy

= S very far awayfrom the crack. Because of the symmetry of the problem, we expect that �

xx

and �yy

to be

1

even functions of both x and y. By symmetry, we only need to look at half of the plate, e.gthe lower half, y 0, as shown below.

On the top surface of the lower half of the plate, we can write down the following boundaryconditions.

�yy

(x, y = 0) ⌘ �py

(x) = 0 , |x| < a

uy

(x, y = 0) ⌘ uy

(x) = 0 , |x| � a(1)

The situation is analogus to the contact problem (flat punch), except that the “contact area”is now in the region of |x| � a. We expect that �

yy

(x, y = 0) / 1/pr for |x| = a + r and

r ! 0. We also expect that �yy

(x, y = 0) ! S for |x| ! 1.Similar to our treatment of the contact problem, we can relate the surface displacement

uy

(x) and the surface traction force py

(x) through the surface Green function.

uy

(x) =

Z+1

�1py

(x0)+ 1

4⇡µlog |x� x0| , �1 < x < +1

duy

(x)

dx=

+ 1

4⇡µ

Z+1

�1

py

(x0)

x� x0 ,

(2)

where + 1/(4µ) = (1� ⌫)/µ in plane strain. Define

g(x) ⌘ 4⇡µ

+ 1

duy

(x)

dx(3)

Then

g(x) =

Z+1

�1

py

(x0)

x� x0 (4)

Because py

(x0) = 0 for |x0| < a, we can limit the integral to the domains (�1, 0] and[0,+1), i.e.,

g(x) =

✓Z �a

�1+

Z+1

a

◆py

(x0)

x� x0 , (5)

By symmetry, we know that duy

(x)/dx = 0 hence g(x) = 0 for |x| > a. Therefore, we aresearching for p

y

(x0) in |x0| > a such that✓Z �a

�1+

Z+1

a

◆py

(x0)

x� x0 = 0 , (6)

This is very similar to the flat punch problem in Lecture Note 10 Contact, except that theregion where p

y

(x) = 0 and the region where duy

(x)/dx = 0 are reversed.

2

2 Stress Field on the Crack Plane

Let us define

g(x) ⌘ 4⇡µ

+ 1

duy

(x)

dx(7)

so that Eq. (5) can be written as

g(x) =

Z �a

�1+

Z+1

a

�py

(x0)

x� x0 , (8)

The solution of this singular integral equation can be written as [1] ,

py

(x) = � 1

⇡2

(x� a)1/2

(x+ a)1/2

Z �a

�1+

Z+1

a

�(x0 + a)1/2

(x0 � a)1/2g(x0)

x� x0 dx0 +

Ax+B

(x+ a)1/2 (x� a)1/2(9)

where A and B are arbitrary constants. In the special case of g(x) = 0 in the domains�1 < x a and a x < 1, the solution can be simplified to (for more details seeAppendix A),

py

(x) =A+B/xp1� (a/x)2

(10)

By symmetry, we expect py

(x) to be an even function of x, i.e. py

(x) = py

(�x). This requiresB = 0. To determine the constant p

0

, we invoke the boundary condition that

limx!1

py

(x) = �S (11)

Therefore, A = �S, so that,

py

(x) = � Sp1� (a/x)2

= � S · |x|px2 � a2

(12)

�yy

(x, y = 0) =S · |x|px2 � a2

(13)

We note that the stress is singular at the crack tips, x = ±a. To extract the nature of thestress singularity, let x = a+ r and take the limit of r ! 0+,

�yy

(x, y = 0) ⇡ S ap2 a r

= S

ra

2

1pr

(14)

3

In Lecture Note 12 Wedge and Notch, we have seen that the singular stress field of a crackunder Mode I loading can be characterized by the stress intensity factor K

I

as,

�rr

=K

Ip2⇡r

✓5

4cos

✓

2� 1

4cos

3✓

2

◆(15)

which corresponds to (when ✓ = 0)

�yy

(x, y = 0) =K

Ip2⇡r

(16)

Therefore, the stress intensity factor for the slit-like crack is

KI

= Sp⇡ a (17)

3 Crack Opening Displacement

Given that py

(x) = �S/p

1� (a/x)2 for |x| > a, we can obtain uy

(x) in the region |x| < aas well. In the region |x| < a,

0 6= duy

(x)

dx=

+ 1

4⇡µ

✓Z �a

�1+

Z+1

a

◆py

(x0)

x� x0 (18)

Definex ⌘ a cos� , x0 ⌘ a

cos ✓(19)

we now havedu

y

(x)

dx=

+ 1

4⇡µ

Z⇡

0

py

�a

cos ✓

�

a cos�� a

cos ✓

a

cos2 ✓sin ✓ d✓

=+ 1

4⇡µ

Z⇡

0

py

�a

cos ✓

�sin ✓

cos� cos ✓ � 1

d✓

cos ✓

(20)

Recall that py

(a/ cos ✓) sin ✓ = �S, we have,

duy

(x)

dx=

+ 1

4⇡µS

Z⇡

0

d✓

(1� cos� cos ✓) cos ✓(21)

We now need another mathematical identity.

1

⇡

Z⇡

0

d✓

(1� cos� cos�) cos ✓=

cos�

sin�(22)

(This can be verified by check angular integral 3.m on Canvas.) Therefore,

duy

(x)

dx=

+ 1

4⇡µS ⇡

cos�

sin�=

+ 1

4µS

x/ap1� (x/a)2

uy

(x) = �+ 1

4µS a

p1� (x/a)2

(23)

4

uy

(x) = �1� ⌫

µS a

p1� (x/a)2 (24)

Therefore, the shape of the crack opening is an ellipse.

d(x) = �2uy

(x) =2(1� ⌫)

µS a

p1� (x/a)2 (25)

4 Enthalpy

A mechanical system subjected to external load will evolve in the direction to reduce itsenthalpy. The stable equilibrium state of the mechanical system under external load hasminimum enthalphy. The enthalpy H is defined as

H = E ��WLM

(26)

where E is the internal energy of the system, and �WLM

is the work done by the loadingmechanism.

To develop intuition about enthalpy, consider a harmonic spring with spring constant ksubjected to a force F as shown below.

The internal energy of the spring is

E =1

2kx2 (27)

5

where x is the displacement of the right end of the spring from its natural (zero force)position, and the left end of the spring is supposed to be fixed. The work done by theexternal force F applied to the right end of the spring is

�WLM

= F x (28)

Therefore, the enthalpy is

H =1

2kx2 � F x (29)

The minimum of H is achieved when

@H

@x= kx� F = 0

x =F

k

(30)

which is what we expect intuitively. It can be seen that when x is at the equilibrium value,the enthalpy in this system is exactly the negative of the energy, i.e. H = �E.

One way to think of enthalpy H is to consider the energy of an extended system thatalso include the energy of the loading mechanism. For example, imagine that the force F isapplied by a weight, F = mg, as shown below.

The total energy of the extended system include the elastic energy of the spring plus thepotential energy of the weight. Let the height of the weight be h = �x. The total energy is

Etot

=1

2kx2 +mgh =

1

2kx2 �mgx =

1

2kx2 � Fx (31)

This is exactly the same as the enthalpy H of the spring subjected to external force F .We now consider a linear elastic medium subjected to traction forces ~T on section S

t

ofits surface, as shown below.

6

The stable equilibrium state of the elastic medium is the state that minimizes the enthalpy

H = E ��WLM

=

Z

⌦

1

2�ij

"ij

dV �Z

St

Tj

uj

dS (32)

We can imagine that the surface tractions are applied by attaching the surface to many (tiny)weights. The enthalpy H is the elastic strain energy E of the medium plus the potentialenergy of all these weights.

To develop intuitions, consider a bar subjected to uniaxial tension with the only non-zerostress component being �A

yy

, as shown below.

7

The elastic energy E, work done �WLM

, and enthalpy H are

E =1

2�A

yy

"yy

V

�WLM

=��A

yy

A�· ("

yy

L) = �A

yy

"yy

V

H =1

2�A

yy

"yy

V � �A

yy

"yy

V = �1

2�A

yy

"yy

V = �E

(33)

In fact, the resultH = �E is generally true for any linear elastic medium that has no internal stresssources, i.e. if the medium contains zero stress field when no external traction is applid.

5 Enthalpy of Crack

Because the system under load evolves towards the state with minimum enthalphy H, andimportant quantity in fracture mechanics is the change of enthalpy �H when a crack isintroduced into the solid.

�H = H1

�H0

(34)

where H0

is the enthalpy of the system in State 0, which does not contain the crack, and H1

is the enthalpy of the system in State 1, which contains the crack. Assume that the systemis subjected to a uniaxial applied stress, �A

yy

= S. Let E0

and E1

be the elastic energy of thesystem in State 0 and State 1, respectively. Then,

H0

= �E0

= �1

2�A

yy

"Ayy

V (35)

H1

= �E1

(36)

Intuitively, we would expect H1

< H0

, �H < 0, i.e. the introduction of a crack lowersthe enthalpy of the system. This reduction of enthalpy provides the driving force for cracknucleation and growth. Since H

1

= �E1

and H0

= �E0

, we should have �E = E1

�E0

> 0.This means that the introduciton of a crack to the solid under load actually increases theelastic energy of the solid.

8

It is possible to obtain H1

by integrating the strain energy density in the entire volumeof the solid (which would give E

1

). Doing so would require the knowledge of the stress fieldin the entire volume of the solid. An alterativel approach is to imagine a reversible processof transforming the system from State 0 (with no crack) to State 1 (with crack). The workdone to the system, �W

0!1

, equals the enthalpy change, �H,

�W0!1

= �H (37)

Let us consider the following two steps of transforming the system from State 0 to State1, as show in the diagram below.

Starting from State 0, we first introduce a cut inside the solid, but apply traction S on thetwo opposing faces of the cut to prevent the cut surfaces from opening up. In this case, thestress field inside the solid remains uniform (�

yy

= S) and the enthalpy remains at H0

. Thetraction on the upper surface is T+

y

= �S and the traction on the lower surface is T�y

= S.In the second step, we gradually reduce the traction forces, T+

y

and T�y

, to zero, until State 1is reached. By the time State 1 is reached, the two surfaces of the crack would have displacedby u+

y

= �uy

(x) and u�y

= uy

(x), where uy

(x) is the surface displacement of the lower faceof the crack we solved earlier.

To obtain �H we need to compute the work done by the forces we applied on the cracksurfaces as we gradually remove our forces and let the crack surfaces open up. Consider theupper surface of the crack. The forces we apply gradually changes from T+

y

= �S to 0 asthe displacement gradually changes from 0 to �u

y

(x), as shown in the diagram below.

9

The work done during this process is

�W+ =

Za

�a

"Z �uy(x)

0

T+

y

du+

y

(x)

#dx

=

Za

�a

1

2(�S) (�u

y

(x)) dx

=1

2S

Za

�a

uy

(x) dx

(38)

The same amount of work is done on the lower surface, i.e. �W� = �W+. Therefore, theenthalpy change for introducing the crack is,

�H = �W0!1

= �W+ +�W� =1

2S

Za

�a

2 uy

(x) dx = �1

2S

Za

�a

d(x) dx (39)

where d(x) is the crack opening displacement,

d(x) =2(1� ⌫)

µS a

r1�

⇣xa

⌘2

(40)

Therefore, the enthalpy change is proportional to the applied stress S times the total areaenclosed in the crack opening.

�H = �1

2S2(1� ⌫)

µS a

r1�

⇣xa

⌘2

= �1� ⌫

µS2 a

⇡

2a

= �1� ⌫

2µS2 ⇡ a2 (plane strain)

(41)

This is the excess enthalpy of a slit-like crack (per unit length in z-direciton) under mode-Iloading. Note:

• �H < 0. Enthalpy is lowered by having the crack in the solid, because it allows theapplied stress �A

yy

to do more work.

10

• �H / S2. Enthalpy change is proportional to the applied stress squared. If there areno applied stress, there is no enthalpy change.

• �H / a2. Larger crack leads to greater enthalpy change under the same applied stress.

6 Driving Force for Crack Extension

Given the excess enthalpy of the crack, we can define the elastic driving force fel

for crackextension (i.e. growth) as the negative slope of the curve when �H is plotted against thecrack length 2a, as shown in the diagram below.

fel

= �@�H

@(2a)= �1

2

@

@a

✓�1� ⌫

2µS2⇡a2

◆

=⇡(1� ⌫)

2µS2 a

(42)

Since KI

= Sp⇡ a, we have

fel

=1� ⌫

2µK2

I

(43)

consistent with the expression promised in Lecture Note 12 Wedge and Notch.We can see that f

el

> 0, meaning that from the point of view of reducing the (elastic)enthalpy, the crack always prefers to grow (i.e. become longer). Since f

el

/ a, the drivingforce for crack extension becomes even bigger as the crack grows longer.

That fel

is positive for any crack of non-zero size may suggest that all cracks are unstableas soon as a stress �A

yy

= S is applied, no matter how small is S. But this is not the case.Even for brittle materials, a critical stress is required for a crack to start growing.

This discrepany was resolved by Gri�th (1921) [2] (see Canvas, Extra Reading). Gri�threcognized that creating a new crack or growing an existing crack requires the creation of newsurfaces, which costs extra energy. Let �

s

be the surface energy (per unit area) of the solid.Define �G as the change of Gibbs free energy of the system (per unit length), includingcontributions from both the (elastic) enthalpy and the surface energy, when introducing thecrack.

�G = �H + �s

· 2 · 2a (44)

where the first factor of 2 corresponds to the upper and lower surfaces of the crack, and 2ais the crack length.

11

�G = �1� ⌫

2µS2 ⇡ a2 + 4 �

s

a (45)

The total driving force for crack extension (now including the surface energies) is

ftot

= � @�G

@(2a)= �1

2

@

@a

✓�1� ⌫

2µS2 ⇡ a2 + 4 �

s

a

◆

=⇡(1� ⌫)

2µS2 a� 2 �

s

(46)

Below a critical size, the driving force for crack extension is negative. Above this critical size,the driving force becomes positive. The critical crack size, 2a

c

can be obtained by settingftot

= 0.⇡(1� ⌫)

2µS2 a� 2 �

s

= 0 (47)

Hence

ac

=4µ

⇡(1� ⌫)

�s

S2

(plane strain) (48)

This means that under a given applied stress �A

yy

= S, cracks with length 2a < 2ac

do notgrow (i.e. they are stable), while cracks with length 2a > 2a

c

are unstable (i.e. leading tofracture).

If we consider a given crack of size 2a, then there exists a critical stress,

Sc

=

s4µ �

s

⇡(1� ⌫)a(plane strain) (49)

such that if �A

yy

< Sc

, the crack does not grow (i.e. it is stable); if �A

yy

> Sc

, the crack isunstable. The above predictions about whether a crack is stable against growth is called theGri�th criterion. It is applicable to brittle materials where the resistance to crack growthcomes mainly from the surface energies of the new crack surfaces.

Gri�th performed many experiments on glass bulbs containing cracks with various sizes.He pressurized the bulbs until they burst. The stress within the walls of the glass bulbsis proportional to the gauge pressure. The measured critical stress at fracture from theseexperiments allowed Gri�th to estimate the surface energy of glass, which agreed well withother independent measurements of surface energy, thus validating the Gri�th criteria of(brittle) fracture.

12

In fact, Gri�th worked under the plane stress assumption (suitable for thin shells of theglass bulb). To obtain corresponding expressions for plane stress, we first express our planestrain expressions in terms of the Kolosov’s constant ( = 3� 4⌫ in plane strain).

d(x) =+ 1

2µS a

p1� (x/a)2

�H = �+ 1

8µS2 ⇡ a2

fel

=⇡ (+ 1)

8µS2 a =

+ 1

8µK2

I

ftot

=⇡ (+ 1)

8µS2 a� 2 �

s

Sc

=

s16µ �

s

⇡(+ 1)a

(50)

In plane stress, = (3� ⌫)(1 + ⌫), + 1 = 4/(1 + ⌫), so that

d(x) =4

2µ(1 + ⌫)S a

p1� (x/a)2

�H = � 1

2µ(1 + ⌫)S2 ⇡ a2 = � 1

ES2 ⇡ a2

fel

=⇡

2µ(1 + ⌫)S2 a =

1

2µ(1 + ⌫)K2

I

=K2

I

E

ftot

=⇡

2µ(1 + ⌫)S2 a� 2 �

s

Sc

=

r4µ(1 + ⌫) �

s

⇡ a=

r2E �

s

⇡ a

(51)

where E = 2µ(1 + ⌫) is Young’s modulus.

13

A Solution of Singular Integral Equation

The book of Mikhlin [1] provides the solution to the singular integral equation of the followingform,

g(x) =

Z

L

f(x0)

x� x0 dx0 (52)

where L is a contour on the complex plane. Here g(x) is known on the contour L, and thegoal is to find f(x) on the contour.

If L is a closed contour, then the solution is

f(x) = � 1

⇡2

Z

L

g(x0)

x� x0 dx0 (53)

In fact, this result is the same as the Hilbert transform (and its inverse) when the contour Lis (�1,+1).

If L is an unclosed continous contour, with ↵ and � being its ending points, then the solutionis

f(x) = � 1

⇡2

✓x� ↵

x� �

◆1/2

Z

L

✓x0 � �

x0 � ↵

◆1/2

g(x0)

x� x0 dx0 +

C

(x� ↵)1/2(x� �)1/2(54)

where C is an arbitrary constant (because the solution is not unique). An alternative formof the solution is

f(x) = � 1

⇡2

1

(x� ↵)1/2(x� �)1/2

Z

L

(x0 � �)1/2(x0 � ↵)1/2g(x0)

x� x0 dx0 +

C 0

(x� ↵)1/2(x� �)1/2

(55)where C 0 is an arbitrary constant. In particular, if the contour L is the region [�c, c] on thereal axis, i.e. ↵ = �c and � = c,

f(x) = � 1

⇡2

1pc2 � x2

Z

L

pc2 � x02 g(x0)

x� x0 dx0 +

C 00pc2 � x2

(56)

where C 00 is an arbitrary constant. We have used this result in Lecture Note 10 Contact.

We now consider the case of unclosed discontinuous contour. If the contour L consists of nsimple arcs, L

1

, L2

, · · · , Ln

, no pair of which has any common point, then the solution ofthe integral equation is

f(x) = � 1

⇡2

nY

k=1

✓x� ↵

k

x� �k

◆1/2

Z

L

nY

k=1

✓x0 � �

k

x0 � ↵k

◆1/2

g(x0)

x� x0 dx0 +

Qn�1

(x)Qn

k=1

(x� ↵k

)1/2(x� �k

)1/2

(57)where ↵

k

and �k

is the starting and ending points of Lk

, respectively, and Qn�1

(x) is anarbitrary polynomial of degree n� 1.

We now apply this to the situation where L consists of two sections on the real axis:(�1,�a] and [a,1). Therefore, n = 2, Q

n�1

(x) = A0x+B0, where A0 and B0 are arbitrary

14

constants. We will let ↵1

= ��, �1

= �a, ↵2

= a, �2

= �, and let � ! 1 at the end ofthe analysis. Because |x| ⌧ � and |x0| ⌧ �, we can replace x � � by ��, and x + � by �.Therefore, the solution of the integral equation becomes,

f(x) = � 1

⇡2

� (x� a)

(x+ a)(��)

�1/2

Z

L

(x0 + a)(��)

� (x0 � a)

�1/2

g(x0)

x� x0 dx0

+Ax+B

�1/2 (x+ a)1/2 (x� a)1/2 (��)1/2

= � 1

⇡2

(x� a)1/2

(x+ a)1/2

Z

L

(x0 + a)1/2

(x0 � a)1/2g(x0)

x� x0 dx0 +

Ax+B

(x+ a)1/2 (x� a)1/2

(58)

where A and B are arbitrary constants.We now consider the special case where g(x) = 0 in the domains �1 < x �a and

a x < 1. Then the solution of the integral equation simply becomes,

f(x) =Ax+B

(x+ a)1/2 (x� a)1/2(59)

where A and B are arbitrary constants. Some care is needed to see exactly how the functionf(x) looks like in the domains x � a and x �a.

For x � a,

f(x) =Ax+Bp

x+ apx� a

=Ax+Bpx2 � a2

(60)

For x �a,

f(x) =Ax+B

ip

�(x+ a) ip

�(x� a)= � Ax+Bp

x2 � a2(61)

Therefore, for |x| � a,

f(x) =(Ax+B) sgn(x)p

x2 � a2(62)

where sgn(x) = 1 if x > 0 and sgn(x) = �1 if x < 0. An alternative way to write thesolution is,

f(x) =A|x|+B sgn(x)p

x2 � a2(63)

or, equivalently,

f(x) =A+B/xp1� (a/x)2

(64)

References

1. S. G. Mikhlin, Integral Equations – and their applications to certain problems in me-chanics, mathematical physics and technology, 2nd ed., Translated from the Russianby A. H. Armstrong, Pergamon Press (1964).

2. A. A. Grifith, The Phenomena of Rupture and Flow in Solids, Philos. Trans. Roy.Soc. Lond. A, 221, 163-198 (1921).

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