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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "elas32.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 12 "groptions();" }}{PARA 6 "" 
1 "" {TEXT -1 31 "`grOptionAlterSize    ` = false" }}{PARA 6 "" 1 "" 
{TEXT -1 30 "`grOptionCoordNames   ` = true" }}{PARA 6 "" 1 "" {TEXT 
-1 27 "`grOptionDefaultSimp  ` = 8" }}{PARA 6 "" 1 "" {TEXT -1 30 "`gr
OptionDisplayLimit ` = 5000" }}{PARA 6 "" 1 "" {TEXT -1 30 "`grOptionL
LSC         ` = true" }}{PARA 6 "" 1 "" {TEXT -1 47 "`grOptionMetricPa
th   ` = `c:/Grtii(5)/Metrics`" }}{PARA 6 "" 1 "" {TEXT -1 42 "`grOpti
onqloadPath    ` = `(not assigned)`" }}{PARA 6 "" 1 "" {TEXT -1 29 "`g
rOptionTermSize     ` = 100" }}{PARA 6 "" 1 "" {TEXT -1 31 "`grOptionT
race        ` = false" }}{PARA 6 "" 1 "" {TEXT -1 30 "`grOptionTimeSta
mp    ` = true" }}{PARA 6 "" 1 "" {TEXT -1 31 "`grOptionVerbose      `
 = false" }}{PARA 6 "" 1 "" {TEXT -1 30 "`grOptionWindows      ` = tru
e" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}{PARA 6 "" 1 "" {TEXT -1 65 "grOp
tionDefaultSimp values: 0=None, 1=simplify, 2=simplify[trig]," }}
{PARA 6 "" 1 "" {TEXT -1 63 "  3=simplify[power] 4=simplify[hypergeom]
, 5=simplify[radical]," }}{PARA 6 "" 1 "" {TEXT -1 57 "  6=expand, 7=f
actor, 8=normal, 9=sort, 10=simplify[sqrt]" }}{PARA 6 "" 1 "" {TEXT 
-1 22 "  11=simplify[trigsin]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 18 "grlib(elasticity):" }}{PARA 6 "" 1 "" {TEXT -1 22 "The Elasticit
y Package" }}{PARA 6 "" 1 "" {TEXT -1 23 "Last Modified May, 1998" }}
{PARA 6 "" 1 "" {TEXT -1 53 "Created by P. Musgrave and K. Lake. Copyr
ight 1996-98" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 94 "Conventions: Deformed state $B$ with metric $G_\{ij\}$ \+
generated from coordinates $(y1,y2,y3)$, " }}{PARA 0 "" 0 "" {TEXT -1 
86 "undeformed state $B_\{0\}$ with metric $g_\{ij\}$ generated from c
oordinates $(x1,x2,x3)$." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 17 "3.2 Simple shear:" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 216 "Take the body coordinates $(x,y,z)$ \+
and consider $B_\{0\}$ the cuboid defined by $(x,y,z) = \\pm\{(a,b,c)
\}$ where $a, b, c$ are constants. Under simple shearing of the cuboid
 we have $(y_\{1\},y_\{2\},y_\{3\}) = (x + Ky,y,z)$." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{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%\"aG-%'vectorG6#7%%#y1G%#y
2G%#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 40 "xform1:=[y1(x,y)=
x+K*y,y2(y)=y,y3(z)=z];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'xform1G7
%/-%#y1G6$%\"xG%\"yG,&F*\"\"\"*&%\"KGF-F+F-F-/-%#y2G6#F+F+/-%#y3G6#%\"
zGF8" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "grtransform(ey,defo
rmed,xform1):" }}{PARA 6 "" 1 "" {TEXT -1 35 "The new default metric i
s: deformed" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "To generate $g_\{i
j\}$:" }}{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%\"aG-%'vectorG6#7%%#x1G%#x
2G%#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 34 "xform2:=[x1(x)=x,x2(y)=y,x3(z)=z];" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%'xform2G7%/-%#x1G6#%\"xGF*/-%#x2G6#%\"yGF//-%#x3
G6#%\"zGF4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "grtransform(e
x,undeformed,xform2);" }}{PARA 6 "" 1 "" {TEXT -1 37 "The new default \+
metric is: undeformed" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "bo
dy(g=undeformed,G=deformed):" }}{PARA 6 "" 1 "" {TEXT -1 36 "The defau
lt 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$\"#g!\"$" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 18 "gralter(_,factor):" }}{PARA 6 "" 1 "" {TEXT 
-1 48 "Component simplification of a GRTensorII object:" }}{PARA 6 "" 
1 "" {TEXT -1 1 " " }}{PARA 6 "" 1 "" {TEXT -1 47 "Applying routine fa
ctor to object strain(dn,dn)" }}{PARA 6 "" 1 "" {TEXT -1 36 "Applying \+
routine factor to object I1" }}{PARA 6 "" 1 "" {TEXT -1 36 "Applying r
outine factor to object I2" }}{PARA 6 "" 1 "" {TEXT -1 36 "Applying ro
utine factor to object I3" }}{PARA 6 "" 1 "" {TEXT -1 42 "Applying rou
tine factor to object B(up,up)" }}{PARA 6 "" 1 "" {TEXT -1 47 "Applyin
g routine factor 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~defor
med~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%.Strain~TensorG" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'strainG6$%#dnGF&" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#/*&&%(strain~G6#%\"aG\"\"\"&%!G6#%\"bGF)-%'matrixG6#
7%7%\"\"!,$%\"KG#F)\"\"#F37%F4,$*$)F5F7\"\"\"F6F37%F3F3F3" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#%8Elasticity~Invariant~I1G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%$I1~G,&\"\"$\"\"\"*$)%\"KG\"\"#\"\"\"F'" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#%8Elasticity~Invariant~I2G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%$I2~G,&\"\"$\"\"\"*$)%\"KG\"\"#\"\"\"F'" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#%8Elasticity~Invariant~I3G" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%$I3~G\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%4Elas
ticity~Tensor~BG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%\"BG6$%#upGF&" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&)%#B~G%\"aG\"\"\")%!G%\"bGF(-%'mat
rixG6#7%7%,&\"\"#F(*$)%\"KGF2\"\"\"F(,$F5!\"\"\"\"!7%F7F2F97%F9F9F1" }
}{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%,,%$PhiGF(%$psiG\"\"#*
&F3F()%\"KGF4\"\"\"F(*&%\"pGF(F6F8F(F:F(,$*&F7F(,&F3F(F:F(F(!\"\"\"\"!
7%F;,(F2F(F3F4F:F(F?7%F?F?,*F2F(F3F4F5F(F:F(" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 115 "Because of the form of the invariants $I$,$\\Phi$, $\\
psi$ and $p$ are constants. $p$ is solved from $\\tau^\{33\} = 0$:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "P:=solve(grcomponent(stress(up,up),
[3,3])=0,p):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "grmap(stres
s(up,up),subs,p=P,`x`):" }}{PARA 6 "" 1 "" {TEXT -1 38 "Applying routi
ne subs to stress(up,up)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 
"grdisplay(stress(up,up)):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%<For~th
e~deformed~spacetime:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%.stress~ten
sorG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'stressG6$%#upGF&" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/*&)%$tauG%\"aG\"\"\")%!G%\"bGF(-%'matrixG6#
7%7%*&,(%$PhiG!\"\"%$psiG!\"#*&F5F()%\"KG\"\"#\"\"\"F4F(F8F;,$*&F9F(,(
F5F4F3F4F7F4F(F4\"\"!7%F<,$F7F4F?7%F?F?F?" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 93 "The normal component of applied force on the side of the \+
cube is given by $\\tau^\{11\}/G^\{11\}$." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "grcomponent(stress(up,up),[1
,1])/grcomponent(g(up,up)[deformed],[1,1]);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#*&*&,(%$PhiG!\"\"%$psiG!\"#*&F(\"\"\")%\"KG\"\"#\"\"\"F
'F+F,F/F/,&*$F,F/F+F+F+!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 114 "
 The tangential component is evaluated from $\\tau^\{1a\}G_\{a2\}/ \\s
urd\{G_\{11\}G_\{22\}\}$. This is calculated as follows:" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "grdef(`L\{^a
 c\}:=stress\{^a ^b\}*g\{b c\}`):" }}{PARA 6 "" 1 "" {TEXT -1 31 "Crea
ted definition for L(up,dn)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
17 "grcalc(L(up,dn)):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%*CPU~Time~G
$\"#5!\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "grcomponent(L(
up,dn),[1,2])/(grcomponent(g(up,up),[1,1])^(1/2)*grcomponent(g(dn,dn),
[2,2])^(1/2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&*&%\"KG\"\"\"%$p
siGF'F'*&F&\"\"\"%$PhiGF'F'F*,&*$)F&\"\"#F*F'F'F'!\"\"" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "3 0 0" 0 }{VIEWOPTS 1 
1 0 1 1 1803 }