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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "elas33.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 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 52 "\{\\bf 3.
3 Pure torsion of a right circular cylinder:\}" }}{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#/%2Defa
ult~spacetimeG%#eyG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%6For~the~ey~sp
acetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%,CoordinatesG" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%\"xG6#%#upG" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/)%#x~G%\"aG-%'vectorG6#7%%#y1G%#y2G%#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 "" {TEXT -1 0 "" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 68 "xform1:=[y1(r,theta)=r*cos(theta),y2(r,theta)=r*
sin(theta),y3(z)=z];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'xform1G7%/-
%#y1G6$%\"rG%&thetaG*&F*\"\"\"-%$cosG6#F+F-/-%#y2GF)*&F*\"\"\"-%$sinGF
0F-/-%#y3G6#%\"zGF<" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "grtr
ansform(ey,deformed,xform1):" }}{PARA 6 "" 1 "" {TEXT -1 35 "The new d
efault metric is: deformed" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
23 "gralter(g(dn,dn),trig):" }}{PARA 6 "" 1 "" {TEXT -1 48 "Component \+
simplification of a GRTensorII object:" }}{PARA 6 "" 1 "" {TEXT -1 1 "
 " }}{PARA 6 "" 1 "" {TEXT -1 52 "Applying routine `simplify[trig]` to
 object g(dn,dn)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"$6
$!\"$" }}}{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$%#dnGF&" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*&&%#g~G6#%\"aG\"\"\"&%!G6#%\"bGF)-%'matrixG6#7%7%F)\"
\"!F37%F3*$)%\"rG\"\"#\"\"\"F37%F3F3F)" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 21 "To generate $g_\{ij\}$:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 10 "qload(ex):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%2Default~spaceti
meG%#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%\"aG-%'vectorG6#7%%#x1G%#x2G%#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 88 "xform2:=[x1(r,theta,z)=
r*cos(theta-delta*z),x2(r,theta,z)=r*sin(theta-delta*z),x3(z)=z];" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'xform2G7%/-%#x1G6%%\"rG%&thetaG%\"z
G*&F*\"\"\"-%$cosG6#,&F+!\"\"*&%&deltaGF.F,F.F.F./-%#x2GF),$*&F*\"\"\"
-%$sinGF1F.F3/-%#x3G6#F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
34 "grtransform(ex,undeformed,xform2);" }}{PARA 6 "" 1 "" {TEXT -1 37 
"The new default metric is: undeformed" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 30 "gralter(g(dn,dn),trig,factor):" }}{PARA 6 "" 1 "" 
{TEXT -1 48 "Component simplification of a GRTensorII object:" }}
{PARA 6 "" 1 "" {TEXT -1 1 " " }}{PARA 6 "" 1 "" {TEXT -1 52 "Applying
 routine `simplify[trig]` to object g(dn,dn)" }}{PARA 6 "" 1 "" {TEXT 
-1 42 "Applying routine factor to object g(dn,dn)" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%*CPU~Time~G$\"#q!\"$" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "grdisplay(_);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%>Fo
r~the~undeformed~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%8Cova
riant~metric~tensorG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"gG6$%#dnGF
&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&&%#g~G6#%\"aG\"\"\"&%!G6#%\"bG
F)-%'matrixG6#7%7%F)\"\"!F37%F3*$)%\"rG\"\"#\"\"\",$*&F6F9%&deltaGF)!
\"\"7%F3F:,&*&F6F9)F<F8F9F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 30 "body(g=undeformed,G=deformed):" }}{PARA 6 "" 1 "" {TEXT -1 36 
"The default metric is now: deformed." }}}{EXCHG {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$\"#]!\"$" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "gralter(_,trig,factor):" }}
{PARA 6 "" 1 "" {TEXT -1 48 "Component simplification of a GRTensorII \+
object:" }}{PARA 6 "" 1 "" {TEXT -1 1 " " }}{PARA 6 "" 1 "" {TEXT -1 
57 "Applying routine `simplify[trig]` to object strain(dn,dn)" }}
{PARA 6 "" 1 "" {TEXT -1 46 "Applying routine `simplify[trig]` to obje
ct I1" }}{PARA 6 "" 1 "" {TEXT -1 46 "Applying routine `simplify[trig]
` to object I2" }}{PARA 6 "" 1 "" {TEXT -1 46 "Applying routine `simpl
ify[trig]` to object I3" }}{PARA 6 "" 1 "" {TEXT -1 52 "Applying routi
ne `simplify[trig]` to object B(up,up)" }}{PARA 6 "" 1 "" {TEXT -1 57 
"Applying routine `simplify[trig]` to object stress(up,up)" }}{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 "Applying routine factor to object I3" }}
{PARA 6 "" 1 "" {TEXT -1 42 "Applying routine factor to object B(up,up
)" }}{PARA 6 "" 1 "" {TEXT -1 47 "Applying routine factor to object st
ress(up,up)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"$]\"!\"
$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "grdisplay(_);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%<For~the~deformed~spacetime:G" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%.Strain~TensorG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'strainG6$%#dnGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
/*&&%(strain~G6#%\"aG\"\"\"&%!G6#%\"bGF)-%'matrixG6#7%7%\"\"!F3F37%F3F
3,$*&)%\"rG\"\"#\"\"\"%&deltaGF)#F)F97%F3F5,$*&F7F:)F;F9F:#!\"\"F9" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%8Elasticity~Invariant~I1G" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/%$I1~G,&\"\"$\"\"\"*&)%\"rG\"\"#\"\"\")%&de
ltaGF+F,F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%8Elasticity~Invariant~I
2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%$I2~G,&\"\"$\"\"\"*&)%\"rG\"\"
#\"\"\")%&deltaGF+F,F'" }}{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%\"aG\"\"\")%!G%\"bGF(-%'matrixG6#7%7%,&\"\"#F(*&)%\"rGF2\"
\"\")%&deltaGF2F6F(\"\"!F97%F9*&F1F6*$)F5\"\"#F6!\"\"F87%F9F8F2" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%.stress~tensorG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%'stressG6$%#upGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
/*&)%$tauG%\"aG\"\"\")%!G%\"bGF(-%'matrixG6#7%7%,*-%$PhiG6#%\"rGF(-%$p
siGF4\"\"#*(F6F()F5F8\"\"\")%&deltaGF8F;F(-%\"pGF4F(\"\"!F@7%F@*&,,*(F
2F(F:F;F<F;F(F2F(F6F8F9F(F>F(F;*$)F5\"\"#F;!\"\"*&F=F(,&F2F(F6F(F(7%F@
FI,(F2F(F6F8F>F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 113 "Because of t
he form of the invariants $I$, $\\Phi$, $\\psi$ and $p$ are functions \+
of $r$. GRTensor recognizes this." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 50 "We now define the vector $T^\{a\} = \\tau
^\{ab\}||-\{b\}." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "grdef(`T\{^b\}:
=stress\{^a ^b;a\}`);" }}{PARA 6 "" 1 "" {TEXT -1 45 "Created a defini
tion for    stress(up,up,cdn)" }}{PARA 6 "" 1 "" {TEXT -1 28 "Created \+
definition for T(up)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "grc
alc(T(up));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"#]!\"$
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "gralter(_,simplify,fact
or):" }}{PARA 6 "" 1 "" {TEXT -1 48 "Component simplification of a GRT
ensorII object:" }}{PARA 6 "" 1 "" {TEXT -1 1 " " }}{PARA 6 "" 1 "" 
{TEXT -1 41 "Applying routine simplify to object T(up)" }}{PARA 6 "" 
1 "" {TEXT -1 39 "Applying routine factor to object T(up)" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G$\"$\"=!\"$" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 59 "The following command writes the derivatives as subs
cripts." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "grmap(_,autoAlias,`x`):
" }}{PARA 6 "" 1 "" {TEXT -1 35 "Applying routine autoAlias to T(up)" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "grdisplay(_);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#%<For~the~deformed~spacetime:G" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#%&T(up)G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"
TG6#%#upG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/)%\"TG%\"aG-%'vectorG6#7
%,.&%$PhiG6#%\"rG\"\"\"&%$psiGF.\"\"#*(F1F0)F/F3\"\"\")%&deltaGF3F6F0*
(-F2F.F0F/F0F7F6F3&%\"pGF.F0*(-F-F.F0F/F6F7F6!\"\"\"\"!F@" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 60 "The equilibrium condition $T^\{a\} = \\ta
u^\{ab\}||_\{b\} = 0 gives" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "grcom
ponent(T(up),[r])=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,.&%$PhiG6#%
\"rG\"\"\"&%$psiGF'\"\"#*(F*F))F(F,\"\"\")%&deltaGF,F/F)*(-F+F'F)F(F)F
0F/F,&%\"pGF'F)*(-F&F'F)F(F/F0F/!\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 39 "\ndsolve(grcomponent(T(up),[r])=0,p(r));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/-%\"pG6#%\"rG,&-%$IntG6$,,&%$PhiGF&!\"\"&%$
psiGF&!\"#*(F0\"\"\")F'\"\"#\"\"\")%&deltaGF6F7F/*(-F1F&F4F'F4F8F7F2*(
-F.F&F4F'F7F8F7F4F'F4%$_C1GF4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}}{MARK "23 1" 0 }{VIEWOPTS 1 1 0 1 1 1803 }