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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "elas34.ms" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 7 "grtw();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%=GRTensorII~Version~1
.70~(R5)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,31~May~1998G" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#%ZDeveloped~by~Peter~Musgrave,~Denis~Pollney
~and~Kayll~LakeG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%DCopyright~1994-1
998~by~the~authors.G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\\oLatest~ver
sion~available~from:~http://astro.queensu.ca/|irgrtensor/G" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#%PDefaults~read~from~c:/Grtii(5)/Lib/grtenso
r.iniG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%4c:/Grtii(5)/MetricsG" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "grOptionTermSize:=0:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "grlib(elasticity):" }}{PARA 
6 "" 1 "" {TEXT -1 22 "The Elasticity Package" }}{PARA 6 "" 1 "" 
{TEXT -1 23 "Last Modified May, 1998" }}{PARA 6 "" 1 "" {TEXT -1 53 "C
reated by P. Musgrave and K. Lake. Copyright 1996-98" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 94 "Conventions: De
formed state $B$ with metric $G_\{ij\}$ generated from coordinates $(y
1,y2,y3)$, " }}{PARA 0 "" 0 "" {TEXT -1 86 "undeformed state $B_\{0\}$
 with metric $g_\{ij\}$ generated from coordinates $(x1,x2,x3)$." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 63 "\{\\bf 3.
4 Extension,inflation,and torsion of a cylindrical tube\}" }}{PARA 0 "
" 0 "" {TEXT -1 41 "Take the body coordinates $(r,\\theta,z)$." }}
{PARA 0 "" 0 "" {TEXT -1 21 "To generate $G_\{ij\}$:" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 10 "qload(ey);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%
2Default~spacetimeG%#eyG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%6For~the~
ey~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,CoordinatesG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"xG6#%#upG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6%/)%#x~G%\"1G%#y1G/)F%%\"2G%#y2G/)F%%\"3G%#y3G" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#%-Line~elementG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*$)%$~dsG\"\"#\"\"\",(*&%#~dG\"\"\")%#y1G%#2~GF,F,*&F+
F()%#y2GF/F,F,*&F+F()%#y3GF/F,F," }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 68 "xform1:=[y1(r,theta)=r*cos(theta),y2(r,theta)=r*sin(t
heta),y3(z)=z];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'xform1G7%/-%#y1G
6$%\"rG%&thetaG*&F*\"\"\"-%$cosG6#F+F-/-%#y2GF)*&F*\"\"\"-%$sinGF0F-/-
%#y3G6#%\"zGF<" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "grtransfo
rm(ey,deformed,xform1):" }}{PARA 6 "" 1 "" {TEXT -1 35 "The new defaul
t metric is: deformed" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "gr
alter(g(dn,dn),trig);" }}{PARA 6 "" 1 "" {TEXT -1 48 "Component simpli
fication of a GRTensorII object:" }}{PARA 6 "" 1 "" {TEXT -1 1 " " }}
{PARA 6 "" 1 "" {TEXT -1 52 "Applying routine `simplify[trig]` to obje
ct g(dn,dn)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"#g!\"$
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "grdisplay(_);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#%<For~the~deformed~spacetime:G" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#%8Covariant~metric~tensorG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%\"gG6$%#dnGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&
%#g~G6#%\"rG\"\"\"&%!GF'F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%#
g~G6#%&thetaG\"\"\"&%!GF'F)*$)%\"rG\"\"#\"\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*&&%#g~G6#%\"zG\"\"\"&%!GF'F)F)" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 120 "To generate $g_\{ij\}$ note that we repace $\\psi$ \+
in Green and Zerna by $\\delta$ since $\\psi$ is a reserved name in Ma
ple." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "qload(ex);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/%2Default~spacetimeG%#exG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%6For~the~ex~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#%,CoordinatesG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"xG6#%#upG
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/)%#x~G%\"1G%#x1G/)F%%\"2G%#x2G/)F
%%\"3G%#x3G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-Line~elementG" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/*$)%$~dsG\"\"#\"\"\",(*&%#~dG\"\"\")%
#x1G%#2~GF,F,*&F+F()%#x2GF/F,F,*&F+F()%#x3GF/F,F," }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 105 "xform2:=[x1(r,theta,z)=r*Q(r)*cos(theta-del
ta*z),x2(r,theta,z)=r*Q(r)*sin(theta-delta*z),x3(z)=z/lambda];" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'xform2G7%/-%#x1G6%%\"rG%&thetaG%\"z
G*(F*\"\"\"-%\"QG6#F*F.-%$cosG6#,&F+!\"\"*&%&deltaGF.F,F.F.F./-%#x2GF)
,$*(F*\"\"\"F/F>-%$sinGF4F.F6/-%#x3G6#F,*&F,F>%'lambdaG!\"\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "grtransform(ex,undeformed,xf
orm2):" }}{PARA 6 "" 1 "" {TEXT -1 37 "The new default metric is: unde
formed" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "gralter(g(dn,dn),
trig);" }}{PARA 6 "" 1 "" {TEXT -1 48 "Component simplification of a G
RTensorII object:" }}{PARA 6 "" 1 "" {TEXT -1 1 " " }}{PARA 6 "" 1 "" 
{TEXT -1 52 "Applying routine `simplify[trig]` to object g(dn,dn)" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"#]!\"$" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "grdisplay(_);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%>For~the~undeformed~spacetime:G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%8Covariant~metric~tensorG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%\"gG6$%#dnGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&
%#g~G6#%\"rG\"\"\"&%!GF'F),(*$)-%\"QGF'\"\"#\"\"\"F)*(F/F)F(F)-%%diffG
6$F/F(F)F1*&)F(F1F2)F4F1F2F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%#
g~G6#%&thetaG\"\"\"&%!GF'F)*&)%\"rG\"\"#\"\"\")-%\"QG6#F.F/F0" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%#g~G6#%&thetaG\"\"\"&%!G6#%\"zGF)
,$*()%\"rG\"\"#\"\"\")-%\"QG6#F1F2F3%&deltaGF)!\"\"" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/*&&%#g~G6#%\"zG\"\"\"&%!GF'F)*&,&**)%\"rG\"\"#\"\"\"
)-%\"QG6#F0F1F2)%&deltaGF1F2)%'lambdaGF1F2F)F)F)F2*$)F:\"\"#F2!\"\"" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "body(g=undeformed,G=deform
ed):" }}{PARA 6 "" 1 "" {TEXT -1 36 "The default metric is now: deform
ed." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "We proceed without the inc
ompressibility condition $I_\{3\} = 0$" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 54 "grcalc(strain(dn,dn),I1,I2,I3,B(up,up),stress(up,up));" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"$g\"!\"$" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "grmap(_,autoAlias,`x`):" }}{PARA 6 
"" 1 "" {TEXT -1 43 "Applying routine autoAlias to strain(dn,dn)" }}
{PARA 6 "" 1 "" {TEXT -1 32 "Applying routine autoAlias to I1" }}
{PARA 6 "" 1 "" {TEXT -1 32 "Applying routine autoAlias to I2" }}
{PARA 6 "" 1 "" {TEXT -1 32 "Applying routine autoAlias to I3" }}
{PARA 6 "" 1 "" {TEXT -1 38 "Applying routine autoAlias to B(up,up)" }
}{PARA 6 "" 1 "" {TEXT -1 43 "Applying routine autoAlias to stress(up,
up)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "gralter(_,simplify,f
actor);" }}{PARA 6 "" 1 "" {TEXT -1 48 "Component simplification of a \+
GRTensorII object:" }}{PARA 6 "" 1 "" {TEXT -1 1 " " }}{PARA 6 "" 1 "
" {TEXT -1 49 "Applying routine simplify to object strain(dn,dn)" }}
{PARA 6 "" 1 "" {TEXT -1 38 "Applying routine simplify to object I1" }
}{PARA 6 "" 1 "" {TEXT -1 38 "Applying routine simplify to object I2" 
}}{PARA 6 "" 1 "" {TEXT -1 38 "Applying routine simplify to object I3
" }}{PARA 6 "" 1 "" {TEXT -1 44 "Applying routine simplify to object B
(up,up)" }}{PARA 6 "" 1 "" {TEXT -1 49 "Applying routine simplify to o
bject stress(up,up)" }}{PARA 6 "" 1 "" {TEXT -1 47 "Applying routine f
actor to object strain(dn,dn)" }}{PARA 6 "" 1 "" {TEXT -1 36 "Applying
 routine factor to object I1" }}{PARA 6 "" 1 "" {TEXT -1 36 "Applying \+
routine factor to object I2" }}{PARA 6 "" 1 "" {TEXT -1 36 "Applying r
outine factor to object I3" }}{PARA 6 "" 1 "" {TEXT -1 42 "Applying ro
utine factor to object B(up,up)" }}{PARA 6 "" 1 "" {TEXT -1 47 "Applyi
ng routine factor to object stress(up,up)" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%*CPU~Time~G$\"%J5!\"$" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "grdisplay(_);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%<Fo
r~the~deformed~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%.Strain
~TensorG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'strainG6$%#dnGF&" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%(strain~G6#%\"rG\"\"\"&%!GF'F),$*
&,(-%\"QGF'F)F)F)*&F(F)&F0F'F)F)F),(F/F)!\"\"F)F1F)F)#F4\"\"#" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%(strain~G6#%&thetaG\"\"\"&%!GF'F)
,$*()%\"rG\"\"#\"\"\",&-%\"QG6#F/F)!\"\"F)F),&F3F)F)F)F)#F6F0" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%(strain~G6#%&thetaG\"\"\"&%!G6#%
\"zGF),$*()%\"rG\"\"#\"\"\")-%\"QG6#F1F2F3%&deltaGF)#F)F2" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#/*&&%(strain~G6#%\"zG\"\"\"&%!GF'F),$*&,(*$)%'l
ambdaG\"\"#\"\"\"!\"\"**)%\"rGF2F3)-%\"QG6#F7F2F3)%&deltaGF2F3F0F3F)F)
F)F3*$)F1\"\"#F3!\"\"#F4F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%8Elasti
city~Invariant~I1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$I1~G*&,4*$)-%
\"QG6#%\"rG\"\"#\"\"\"F-**)F,F-F.)F)\"\"%F.)%&deltaGF-F.)%'lambdaGF-F.
\"\"\"*,)F,\"\"$F.)F)F:F.F3F.F5F.&F*F+F7F-*,)F,F2F.F(F.F3F.F5F.)F<F-F.
F7*(F)F7F,F7F<F.F-*&F0F.F?F.F7*&F5F.F1F.F7**F5F.F;F.F,F.F<F.F-**F5F.F(
F.F0F.F?F.F7F.*&),&F)F7*&F,F.F<F.F7\"\"#F.)F)\"\"#F.!\"\"" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#%8Elasticity~Invariant~I2G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%$I2~G*&,,*&)%'lambdaG\"\"#\"\"\")-%\"QG6#%\"rGF*F+F**
*F(F+F-\"\"\"F0F2&F.F/F2F**(F(F+)F0F*F+)F3F*F+F2**F5F+F,F+)%&deltaGF*F
+F(F+F2F2F2F+*&),&F-F2*&F0F+F3F+F2\"\"#F+)F-\"\"#F+!\"\"" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#%8Elasticity~Invariant~I3G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%$I3~G*&*$)%'lambdaG\"\"#\"\"\"F**&),&-%\"QG6#%\"rG\"
\"\"*&F1F2&F/F0F2F2\"\"#F*)F.\"\"#F*!\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%4Elasticity~Tensor~BG" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#-%\"BG6$%#upGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%#B~G%\"rG\"
\"\")%!GF'F(*&,(**)F'\"\"#\"\"\")-%\"QG6#F'F/F0)%&deltaGF/F0)%'lambdaG
F/F0F(F(F(*&F7F0F1F0F(F0*&),&F2F(*&F'F(&F3F4F(F(\"\"#F0)F2\"\"#F0!\"\"
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%#B~G%&thetaG\"\"\")%!GF'F(*&,
,*&)%'lambdaG\"\"#\"\"\")-%\"QG6#%\"rGF0F1F(**)F6F0F1F2F1)%&deltaGF0F1
F.F1F(**F.F1F3F(F6F(&F4F5F(F0*(F.F1F8F1)F<F0F1F(F(F(F1*()F6\"\"#F1)F3
\"\"#F1),&F3F(*&F6F1F<F1F(\"\"#F1!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/*&)%#B~G%&thetaG\"\"\")%!G%\"zGF(*&*&%&deltaGF()%'lambdaG\"\"#
\"\"\"F2*$),&-%\"QG6#%\"rGF(*&F9F(&F7F8F(F(\"\"#F2!\"\"" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/*&)%#B~G%\"zG\"\"\")%!GF'F(*&*&)%'lambdaG\"\"#
\"\"\",(*$)-%\"QG6#%\"rGF/F0F/*(F4F(F7F(&F5F6F(F/*&)F7F/F0)F9F/F0F(F(F
0*&),&F4F(*&F7F0F9F0F(\"\"#F0)F4\"\"#F0!\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%.stress~tensorG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'
stressG6$%#upGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%$tauG%\"rG\"
\"\")%!GF'F(*&,0*&-%$PhiG6#F'F()-%\"QGF0\"\"#\"\"\"F(*,-%$psiGF0F()F'F
4F5F1F5)%&deltaGF4F5)%'lambdaGF4F5F(F7F(*(F7F5F<F5F1F5F(*&-%\"pGF0F()F
2\"\"%F5F(**F@F5)F2\"\"$F5F'F(&F3F0F(F4**F@F5F1F5F9F5)FGF4F5F(F5*&),&F
2F(*&F'F5FGF5F(\"\"#F5)F2\"\"#F5!\"\"" }}{PARA 12 "" 1 "" {XPPMATH 20 
"6#/*&)%$tauG%&thetaG\"\"\")%!GF'F(*&,>*,-%$PhiG6#%\"rGF()F1\"\"#\"\"
\")-%\"QGF0\"\"%F4)%&deltaGF3F4)%'lambdaGF3F4F(*.F.F4)F1\"\"$F4)F6F?F4
F9F4F;F4&F7F0F(F3*.F.F4)F1F8F4)F6F3F4F9F4F;F4)FAF3F4F(*&F.F4FDF4F(**F.
F4F6F(F1F(FAF4F3*(F.F4F2F4FEF4F(*(-%$psiGF0F(F;F4FDF4F(*,FJF4F2F4FDF4F
9F4F;F4F(*,FJF4F;F4F6F4F1F4FAF4F3**FJF4F;F4F2F4FEF4F(FJF(*&-%\"pGF0F(F
5F4F(**FPF4F@F4F1F4FAF4F3**FPF4FDF4F2F4FEF4F(F4*()F1\"\"#F4)F6\"\"#F4)
,&F6F(*&F1F4FAF4F(\"\"#F4!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)
%$tauG%&thetaG\"\"\")%!G%\"zGF(*&*(%&deltaGF()%'lambdaG\"\"#\"\"\",**&
-%$PhiG6#%\"rGF()-%\"QGF7F1F2F(**F5F2F:F(F8F(&F;F7F(F1*(F5F2)F8F1F2)F=
F1F2F(-%$psiGF7F(F(F2*$),&F:F(*&F8F2F=F2F(\"\"#F2!\"\"" }}{PARA 12 "" 
1 "" {XPPMATH 20 "6#/*&)%$tauG%\"zG\"\"\")%!GF'F(*&,4*(-%$PhiG6#%\"rGF
()%'lambdaG\"\"#\"\"\")-%\"QGF0\"\"%F5F(*,F.F5F2F5)F7\"\"$F5F1F(&F8F0F
(F4*,F.F5F2F5)F7F4F5)F1F4F5)F=F4F5F(*(-%$psiGF0F(F2F5F?F5F4*,FCF5F2F5F
7F(F1F5F=F5F4**FCF5F2F5F@F5FAF5F(*&-%\"pGF0F(F6F5F(**FHF5F;F5F1F5F=F5F
4**FHF5F?F5F@F5FAF5F(F5*&),&F7F(*&F1F5F=F5F(\"\"#F5)F7\"\"#F5!\"\"" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "Now the incompressibility conditi
on $I_\{3\} = 1$ can be solved explicitly for $Q(r)$" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 34 "dsolve(grcomponent(I3,[])=1,Q(r));" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6(/,&-%#lnG6#%\"rG\"\"\"%$_C1G!\"\"\"\"!/-%\"QGF'*
&*$-%%sqrtG6#,&-%$expG6#,$F*\"\"#F+*&%'lambdaGF))F(F:\"\"\"F)F>F>F(!\"
\"/F.,$F0F+F#/F.*&*$-F36#,&F6F)F;F+F>F>F(F?/F.,$FCF+" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 100 "Following Green and Zerna we first removed der
ivatives of $Q(r)$ and leave $Q(r)$ in the components." }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 121 "grmap(g(dn,dn)[undeformed],strain(dn,dn),I1,I
2,I3,B(up,up),stress(up,up),subs,diff(Q(r),r)=(lambda-Q(r)^2)/(Q(r)*r)
,`x`);" }}{PARA 6 "" 1 "" {TEXT -1 33 "Applying routine subs to g(dn,d
n)" }}{PARA 6 "" 1 "" {TEXT -1 38 "Applying routine subs to strain(dn,
dn)" }}{PARA 6 "" 1 "" {TEXT -1 27 "Applying routine subs to I1" }}
{PARA 6 "" 1 "" {TEXT -1 27 "Applying routine subs to I2" }}{PARA 6 "
" 1 "" {TEXT -1 27 "Applying routine subs to I3" }}{PARA 6 "" 1 "" 
{TEXT -1 33 "Applying routine subs to B(up,up)" }}{PARA 6 "" 1 "" 
{TEXT -1 38 "Applying routine subs to stress(up,up)" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 27 "gralter(_,simplify,expand);" }}{PARA 6 "
" 1 "" {TEXT -1 48 "Component simplification of a GRTensorII object:" 
}}{PARA 6 "" 1 "" {TEXT -1 1 " " }}{PARA 6 "" 1 "" {TEXT -1 44 "Applyi
ng routine simplify to object g(dn,dn)" }}{PARA 6 "" 1 "" {TEXT -1 49 
"Applying routine simplify to object strain(dn,dn)" }}{PARA 6 "" 1 "" 
{TEXT -1 38 "Applying routine simplify to object I1" }}{PARA 6 "" 1 "
" {TEXT -1 38 "Applying routine simplify to object I2" }}{PARA 6 "" 1 
"" {TEXT -1 38 "Applying routine simplify to object I3" }}{PARA 6 "" 
1 "" {TEXT -1 44 "Applying routine simplify to object B(up,up)" }}
{PARA 6 "" 1 "" {TEXT -1 49 "Applying routine simplify to object stres
s(up,up)" }}{PARA 6 "" 1 "" {TEXT -1 42 "Applying routine expand to ob
ject g(dn,dn)" }}{PARA 6 "" 1 "" {TEXT -1 47 "Applying routine expand \+
to object strain(dn,dn)" }}{PARA 6 "" 1 "" {TEXT -1 36 "Applying routi
ne expand to object I1" }}{PARA 6 "" 1 "" {TEXT -1 36 "Applying routin
e expand to object I2" }}{PARA 6 "" 1 "" {TEXT -1 36 "Applying routine
 expand to object I3" }}{PARA 6 "" 1 "" {TEXT -1 42 "Applying routine \+
expand to object B(up,up)" }}{PARA 6 "" 1 "" {TEXT -1 47 "Applying rou
tine expand to object stress(up,up)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#/%*CPU~Time~G$\"$r&!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 
"grdisplay(_);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%>For~the~undeformed
~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%8Covariant~metric~ten
sorG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"gG6$%#dnGF&" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/*&&%#g~G6#%\"rG\"\"\"&%!GF'F)*&*$)%'lambdaG\"\"
#\"\"\"F1*$)-%\"QGF'\"\"#F1!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*
&&%#g~G6#%&thetaG\"\"\"&%!GF'F)*&)%\"rG\"\"#\"\"\")-%\"QG6#F.F/F0" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%#g~G6#%&thetaG\"\"\"&%!G6#%\"zGF)
,$*()%\"rG\"\"#\"\"\")-%\"QG6#F1F2F3%&deltaGF)!\"\"" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/*&&%#g~G6#%\"zG\"\"\"&%!GF'F),&*()%\"rG\"\"#\"\"\")-
%\"QG6#F/F0F1)%&deltaGF0F1F)*&F1F1*$)%'lambdaG\"\"#F1!\"\"F)" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#%<For~the~deformed~spacetime:G" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#%.Strain~TensorG" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%'strainG6$%#dnGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%(str
ain~G6#%\"rG\"\"\"&%!GF'F),&#F)\"\"#F)*&*$)%'lambdaGF.\"\"\"F3*$)-%\"Q
GF'\"\"#F3!\"\"#!\"\"F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%(strai
n~G6#%&thetaG\"\"\"&%!GF'F),&*&)%\"rG\"\"#\"\"\")-%\"QG6#F/F0F1#!\"\"F
0*$F.F1#F)F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%(strain~G6#%&thet
aG\"\"\"&%!G6#%\"zGF),$*()%\"rG\"\"#\"\"\")-%\"QG6#F1F2F3%&deltaGF)#F)
F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%(strain~G6#%\"zG\"\"\"&%!GF
'F),(#F)\"\"#F)*()%\"rGF.\"\"\")-%\"QG6#F1F.F2)%&deltaGF.F2#!\"\"F.*&F
2F2*$)%'lambdaG\"\"#F2!\"\"F9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%8Ela
sticity~Invariant~I1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$I1~G,**()%
'lambdaG\"\"#\"\"\")%\"rGF)F*)%&deltaGF)F*\"\"\"*&F*F**$)-%\"QG6#F,\"
\"#F*!\"\"F/*&*$)F3F)F*F**$)F(\"\"#F*F7F/*$F'F*F/" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%8Elasticity~Invariant~I2G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%$I2~G,**$)-%\"QG6#%\"rG\"\"#\"\"\"\"\"\"*&*$)%'lambda
GF,F-F-*$)F(\"\"#F-!\"\"F.*()F+F,F-F'F-)%&deltaGF,F-F.*&F-F-*$)F2\"\"#
F-F6F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%8Elasticity~Invariant~I3G" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$I3~G\"\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%4Elasticity~Tensor~BG" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#-%\"BG6$%#upGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%#B~G%\"rG\"
\"\")%!GF'F(,(*()F'\"\"#\"\"\")-%\"QG6#F'F.F/)%&deltaGF.F/F(*&F/F/*$)%
'lambdaG\"\"#F/!\"\"F(*$F0F/F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)
%#B~G%&thetaG\"\"\")%!GF'F(,(*&)-%\"QG6#%\"rG\"\"#\"\"\")%&deltaGF2F3F
(*&*$)%'lambdaGF2F3F3*&)F.\"\"#F3)F1\"\"#F3!\"\"F(*&F3F3*&)F1\"\"#F3)F
9\"\"#F3F?F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%#B~G%&thetaG\"\"
\")%!G%\"zGF(*&%&deltaGF()-%\"QG6#%\"rG\"\"#\"\"\"" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#/*&)%#B~G%\"zG\"\"\")%!GF'F(,&*&*$)%'lambdaG\"\"#\"\"
\"F1*$)-%\"QG6#%\"rG\"\"#F1!\"\"F(*$)F4F0F1F(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%.stress~tensorG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'
stressG6$%#upGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%$tauG%\"rG\"
\"\")%!GF'F(,,*&*&-%$PhiG6#F'F()-%\"QGF0\"\"#\"\"\"F5*$)%'lambdaG\"\"#
F5!\"\"F(**-%$psiGF0F()F'F4F5F1F5)%&deltaGF4F5F(*&F<F5*$)F8\"\"#F5F:F(
*&F<F5F1F5F(-%\"pGF0F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%$tauG%&
thetaG\"\"\")%!GF'F(,.*&*&)%'lambdaG\"\"#\"\"\"-%$psiG6#%\"rGF(F1*&)-%
\"QGF4\"\"#F1)F5\"\"#F1!\"\"F(*&-%$PhiGF4F1*&)F8\"\"#F1)F5\"\"#F1F=F(*
(F.F1F?F()%&deltaGF0F1F(*()F8F0F1F2F1FGF1F(*&-%\"pGF4F1*$)F5\"\"#F1F=F
(*&F2F1*&)F5\"\"#F1)F/\"\"#F1F=F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/
*&)%$tauG%&thetaG\"\"\")%!G%\"zGF(,&*(%&deltaGF(-%$PhiG6#%\"rGF()%'lam
bdaG\"\"#\"\"\"F(*(F.F6-%$psiGF1F()-%\"QGF1F5F6F(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*&)%$tauG%\"zG\"\"\")%!GF'F(,**&-%$PhiG6#%\"rGF()%'lam
bdaG\"\"#\"\"\"F(*&-%$psiGF/F()-%\"QGF/F3F4F(*&*&F6F4F1F4F4*$)F9\"\"#F
4!\"\"F(-%\"pGF/F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "We now defi
ne the vector $T^\{a\} = \\tau^\{ab\}||-\{b\}$." }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "grdef(`T\{^b\}:=stress\{^a^b;a\}`):" }}{PARA 6 "" 1 "
" {TEXT -1 45 "Created a definition for    stress(up,up,cdn)" }}{PARA 
6 "" 1 "" {TEXT -1 28 "Created definition for T(up)" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 14 "grcalc(T(up));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%*CPU~Time~G$\"#r!\"$" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "grmap(_,autoAlias,`x`);" }}{PARA 6 "" 1 "" {TEXT -1 
35 "Applying routine autoAlias to T(up)" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "gralter(_,expand):" }}{PARA 6 "" 1 "" {TEXT -1 48 "Co
mponent simplification of a GRTensorII object:" }}{PARA 6 "" 1 "" 
{TEXT -1 1 " " }}{PARA 6 "" 1 "" {TEXT -1 39 "Applying routine expand \+
to object T(up)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"#5!
\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "grdisplay(_);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%<For~the~deformed~spacetime:G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%&T(up)G" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%\"TG6#%#upG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/)%\"TG%\"rG,>*
&*&)-%\"QG6#F&\"\"#\"\"\"&%$PhiGF-\"\"\"F/*$)%'lambdaG\"\"#F/!\"\"F2*&
*(F+F2-F1F-F2&F,F-F2F/*$)F5\"\"#F/F7F.**)F&F.F/F*F/&%$psiGF-F2)%&delta
GF.F/F2**F&F2F*F/-FBF-F2FCF/F.*,F@F/F+F/FFF/FCF/F;F/F.*&FAF/*$)F5\"\"#
F/F7F2*&F*F/FAF/F2*(F+F/FFF/F;F/F.&%\"pGF-F2*&*&F*F/F:F/F/*&F&\"\"\")F
5\"\"#F/F7F2*&*&F*F/FFF/F/F&F7F2*&*&)F5F.F/FFF/F/*&F&\"\"\")F+\"\"#F/F
7!\"\"*&F:F/*&F&\"\"\")F+\"\"#F/F7Fin**F&F/FZF/F:F/FCF/Fin" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 61 "The equilibrium condition $T^\{a\} = \\ta
u^\{ab\}||_\{b\} = 0$ gives" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "expa
nd(dsolve(grcomponent(T(up),[r])=0,p(r)));" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#/-%\"pG6#%\"rG,&*&-%$IntG6$,<*&&%$PhiGF&\"\"\")-%\"QGF&
\"\"#\"\"\"!\"\"*(-F0F&F1F3F1&F4F&F1!\"#*,&%$psiGF&F1)F'F5F6F2F6)%&del
taGF5F6)%'lambdaGF5F6F7*,-F>F&F1F'F1F2F6F@F6FBF6F;*.FEF6F?F6F3F6F@F6F:
F6FBF6F;F=F7*(F=F6F2F6FBF6F7**FEF6F3F6F:F6FBF6F;*&*&F2F6F9F6F6F'!\"\"F
7*&*(F2F6FEF6FBF6F6F'FKF7*&*&)FC\"\"%F6FEF6F6*&)F3\"\"#F6F'\"\"\"FKF1*
&*&F9F6FBF6F6*&)F3\"\"#F6F'\"\"\"FKF1**F'F6FPF6F9F6F@F6F1F'F6*$)FC\"\"
#F6FKF1%$_C1GF1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 96 "Because we hav
e solved for $Q(r)$ we can go back and write the entire analysis in ex
plicit form." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 119 "grmap(g(dn,dn)[und
eformed],strain(dn,dn),I1,I2,I3,B(up,up),stress(up,up),T(up),subs,Q(r)
=(lambda*(r^2+K))^(1/2)/r,`x`);" }}{PARA 6 "" 1 "" {TEXT -1 33 "Applyi
ng routine subs to g(dn,dn)" }}{PARA 6 "" 1 "" {TEXT -1 38 "Applying r
outine subs to strain(dn,dn)" }}{PARA 6 "" 1 "" {TEXT -1 27 "Applying \+
routine subs to I1" }}{PARA 6 "" 1 "" {TEXT -1 27 "Applying routine su
bs to I2" }}{PARA 6 "" 1 "" {TEXT -1 27 "Applying routine subs to I3" 
}}{PARA 6 "" 1 "" {TEXT -1 33 "Applying routine subs to B(up,up)" }}
{PARA 6 "" 1 "" {TEXT -1 38 "Applying routine subs to stress(up,up)" }
}{PARA 6 "" 1 "" {TEXT -1 30 "Applying routine subs to T(up)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "gralter(_,simplify,factor);
" }}{PARA 6 "" 1 "" {TEXT -1 48 "Component simplification of a GRTenso
rII object:" }}{PARA 6 "" 1 "" {TEXT -1 1 " " }}{PARA 6 "" 1 "" {TEXT 
-1 44 "Applying routine simplify to object g(dn,dn)" }}{PARA 6 "" 1 "
" {TEXT -1 49 "Applying routine simplify to object strain(dn,dn)" }}
{PARA 6 "" 1 "" {TEXT -1 38 "Applying routine simplify to object I1" }
}{PARA 6 "" 1 "" {TEXT -1 38 "Applying routine simplify to object I2" 
}}{PARA 6 "" 1 "" {TEXT -1 38 "Applying routine simplify to object I3
" }}{PARA 6 "" 1 "" {TEXT -1 44 "Applying routine simplify to object B
(up,up)" }}{PARA 6 "" 1 "" {TEXT -1 49 "Applying routine simplify to o
bject stress(up,up)" }}{PARA 6 "" 1 "" {TEXT -1 41 "Applying routine s
implify to object T(up)" }}{PARA 6 "" 1 "" {TEXT -1 42 "Applying routi
ne factor to object g(dn,dn)" }}{PARA 6 "" 1 "" {TEXT -1 47 "Applying \+
routine factor to object strain(dn,dn)" }}{PARA 6 "" 1 "" {TEXT -1 36 
"Applying routine factor to object I1" }}{PARA 6 "" 1 "" {TEXT -1 36 "
Applying routine factor to object I2" }}{PARA 6 "" 1 "" {TEXT -1 36 "A
pplying routine factor to object I3" }}{PARA 6 "" 1 "" {TEXT -1 42 "Ap
plying routine factor to object B(up,up)" }}{PARA 6 "" 1 "" {TEXT -1 
47 "Applying routine factor to object stress(up,up)" }}{PARA 6 "" 1 "
" {TEXT -1 39 "Applying routine factor to object T(up)" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"$s)!\"$" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 13 "grdisplay(_);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%
>For~the~undeformed~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%8C
ovariant~metric~tensorG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"gG6$%#d
nGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%#g~G6#%\"rG\"\"\"&%!GF'F)
*&*&%'lambdaGF))F(\"\"#\"\"\"F1,&*$F/F1F)%\"KGF)!\"\"" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/*&&%#g~G6#%&thetaG\"\"\"&%!GF'F)*&%'lambdaGF),&*$
)%\"rG\"\"#\"\"\"F)%\"KGF)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%#
g~G6#%&thetaG\"\"\"&%!G6#%\"zGF),$*(%'lambdaGF),&*$)%\"rG\"\"#\"\"\"F)
%\"KGF)F)%&deltaGF)!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%#g~G6
#%\"zG\"\"\"&%!GF'F)*&,(*()%'lambdaG\"\"$\"\"\")%&deltaG\"\"#F2)%\"rGF
5F2F)*(F/F2F3F2%\"KGF)F)F)F)F2*$)F0\"\"#F2!\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%<For~the~deformed~spacetime:G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%.Strain~TensorG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'
strainG6$%#dnGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%(strain~G6#%
\"rG\"\"\"&%!GF'F),$*&,(*$)F(\"\"#\"\"\"!\"\"%\"KGF3*&%'lambdaGF)F0F2F
)F2,&F/F)F4F)!\"\"#F3F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%(strai
n~G6#%&thetaG\"\"\"&%!GF'F),(*&%'lambdaGF))%\"rG\"\"#\"\"\"#!\"\"F1*&F
.F2%\"KGF)F3*$F/F2#F)F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%(strai
n~G6#%&thetaG\"\"\"&%!G6#%\"zGF),$*(%'lambdaGF),&*$)%\"rG\"\"#\"\"\"F)
%\"KGF)F)%&deltaGF)#F)F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%(stra
in~G6#%\"zG\"\"\"&%!GF'F),$*&,**$)%'lambdaG\"\"#\"\"\"!\"\"*()F1\"\"$F
3)%&deltaGF2F3)%\"rGF2F3F)*(F6F3F8F3%\"KGF)F)F)F)F3*$)F1\"\"#F3!\"\"#F
4F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%8Elasticity~Invariant~I1G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$I1~G*&,0*()%'lambdaG\"\"$\"\"\")%\"
rG\"\"'F+)%&deltaG\"\"#F+\"\"\"**F(F+)F-\"\"%F+F/F+%\"KGF2F2*$F4F+F1*&
)F-F1F+F6F+F1*$)F6F1F+F2*&F(F+F4F+F2*(F(F+F9F+F6F+F2F+*(F)\"\"\",&*$F9
F+F2F6F2\"\"\")F-\"\"#F+!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%8Ela
sticity~Invariant~I2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$I2~G*&,2*&
)%'lambdaG\"\"$\"\"\")%\"rG\"\"%F+\"\"#*(F(F+)F-F/F+%\"KG\"\"\"F/*&F(F
+)F2F/F+F3*(F(F+)F-\"\"'F+)%&deltaGF/F+F3**F(F+F,F+F9F+F2F+F/**F(F+F9F
+F1F+F5F+F3*$F,F+F3*&F1F+F2F+F3F+*()F-\"\"#F+,&*$F1F+F3F2F3\"\"\")F)\"
\"#F+!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%8Elasticity~Invariant~I
3G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$I3~G\"\"\"" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#%4Elasticity~Tensor~BG" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#-%\"BG6$%#upGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%#B~G%\"r
G\"\"\")%!GF'F(*&,,*()%'lambdaG\"\"$\"\"\")F'\"\"%F1)%&deltaG\"\"#F1F(
**F.F1F4F1)F'F6F1%\"KGF(F(*$F8F1F(*&F.F1F8F1F(*&F.F1F9F1F(F1*&)F/\"\"#
F1)F'\"\"#F1!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%#B~G%&thetaG
\"\"\")%!GF'F(*&,.*()%'lambdaG\"\"$\"\"\")%\"rG\"\"%F1)%&deltaG\"\"#F1
F(**F.F1F5F1)F3F7F1%\"KGF(F7*(F.F1F5F1)F:F7F1F(*&F.F1F9F1F(*$F9F1F(F:F
(F1*()F3\"\"#F1,&F>F(F:F(\"\"\")F/\"\"#F1!\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*&)%#B~G%&thetaG\"\"\")%!G%\"zGF(*&*(%&deltaGF(%'lambd
aGF(,&*$)%\"rG\"\"#\"\"\"F(%\"KGF(F(F5*$)F3\"\"#F5!\"\"" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/*&)%#B~G%\"zG\"\"\")%!GF'F(*&*&%'lambdaGF(,(*$)
%\"KG\"\"#\"\"\"F(*$)%\"rG\"\"%F3F2*&)F6F2F3F1F(F2F(F3*&)F6\"\"#F3,&*$
F9F3F(F1F(\"\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%.stress~tens
orG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'stressG6$%#upGF&" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/*&)%$tauG%\"rG\"\"\")%!GF'F(*&,2*(-%$PhiG6#
F'F(%'lambdaGF()F'\"\"#\"\"\"F(*(F.F4F1F4%\"KGF(F(**-%$psiGF0F()F1\"\"
$F4)%&deltaGF3F4)F'\"\"%F4F(*,F8F4F:F4F<F4F2F4F6F4F(*&F8F4F2F4F(*(F8F4
F:F4F2F4F(*(F8F4F:F4F6F4F(*(-%\"pGF0F()F1F3F4F2F4F(F4*&)F1\"\"#F4)F'\"
\"#F4!\"\"" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/*&)%$tauG%&thetaG\"\"\"
)%!GF'F(*&,8*(-%$psiG6#%\"rGF()%'lambdaG\"\"$\"\"\")F1\"\"#F5F(*(-%$Ph
iGF0F(F3F(F6F5F(**)F3\"\"%F5F9F5)%&deltaGF7F5)F1F=F5F(*,F<F5F9F5F>F5F6
F5%\"KGF(F(**F.F5F2F5F>F5F@F5F(*,F.F5F2F5F>F5F6F5FBF5F7**F.F5F2F5F>F5)
FBF7F5F(*(-%\"pGF0F()F3F7F5F6F5F(*(FHF5FJF5FBF5F(*&F.F5F6F5F(*&F.F5FBF
5F(F5*()F1\"\"#F5,&*$F6F5F(FBF(\"\"\")F3\"\"#F5!\"\"" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#/*&)%$tauG%&thetaG\"\"\")%!G%\"zGF(*&*(%&deltaGF(%'l
ambdaG\"\"\",(*(-%$PhiG6#%\"rGF(F/F()F6\"\"#F0F(*&-%$psiGF5F(F7F0F(*&F
:F0%\"KGF(F(F(F0*$)F6\"\"#F0!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/
*&)%$tauG%\"zG\"\"\")%!GF'F(*&,0*(-%$PhiG6#%\"rGF()%'lambdaG\"\"#\"\"
\")F1\"\"%F5F(**F.F5F2F5)F1F4F5%\"KGF(F(*(F3F(F6F5-%$psiGF0F(F4**F3F5F
<F5F9F5F:F5F4*(F3F5F<F5)F:F4F5F(*&-%\"pGF0F(F6F5F(*(FBF5F9F5F:F5F(F5*&
)F1\"\"#F5,&*$F9F5F(F:F(\"\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#%&T(up)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"TG6#%#upG" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#/)%\"TG%\"rG,$*&,H**&%$PhiG6#F&\"\"\"%'lambd
aGF.)F&\"\"$\"\"\"%\"KGF.!\"#*(F+F2F/F2)F&\"\"&F2!\"\"*(&%$psiGF-F.F0F
2F3F2F8*()F/F1F2F:F2F6F2F8*(&%\"pGF-F.)F/\"\"#F2F6F2F8*&F:F2F6F2F8**F+
F2F/F2F&F.)F3FBF2F8**F=F2F:F2)%&deltaGFBF2)F&\"\"(F2F8**-F;F-F.F=F2FGF
2)F&\"\"'F2F4**F=F2F:F2F0F2F3F2F4**F=F2F:F2F&F2FEF2F8**F?F2FAF2F0F2F3F
2F8**)F/\"\"%F2-F,F-F.FGF2FMF2F.*,FSF2FUF2FGF2)F&FTF2F3F2F.*,F=F2F:F2F
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