elas32.ms

> restart:

> grtw();

> groptions();

`grOptionAlterSize ` = false

`grOptionCoordNames ` = true

`grOptionDefaultSimp ` = 8

`grOptionDisplayLimit ` = 5000

`grOptionLLSC ` = true

`grOptionMetricPath ` = `c:/Grtii(5)/Metrics`

`grOptionqloadPath ` = `(not assigned)`

`grOptionTermSize ` = 100

`grOptionTrace ` = false

`grOptionTimeStamp ` = true

`grOptionVerbose ` = false

`grOptionWindows ` = true

grOptionDefaultSimp values: 0=None, 1=simplify, 2=simplify[trig],

3=simplify[power] 4=simplify[hypergeom], 5=simplify[radical],

6=expand, 7=factor, 8=normal, 9=sort, 10=simplify[sqrt]

11=simplify[trigsin]

> grlib(elasticity):

The Elasticity Package

Last Modified May, 1998

Created by P. Musgrave and K. Lake. Copyright 1996-98

Conventions: Deformed state $B$ with metric $G_{ij}$ generated from coordinates $(y1,y2,y3)$,

undeformed state $B_{0}$ with metric $g_{ij}$ generated from coordinates $(x1,x2,x3)$.

3.2 Simple shear:

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 $(y_{1},y_{2},y_{3}) = (x + Ky,y,z)$.

To generate $G_{ij}$:

> qload(ey):

> xform1:=[y1(x,y)=x+K*y,y2(y)=y,y3(z)=z];

> grtransform(ey,deformed,xform1):

The new default metric is: deformed

To generate $g_{ij}$:

> qload(ex):

> xform2:=[x1(x)=x,x2(y)=y,x3(z)=z];

> grtransform(ex,undeformed,xform2);

The new default metric is: undeformed

> body(g=undeformed,G=deformed):

The default metric is now: deformed.

> grcalc(strain(dn,dn),I1,I2,I3,B(up,up),stress(up,up));

> gralter(_,factor):

Component simplification of a GRTensorII object:

Applying routine factor to object strain(dn,dn)

Applying routine factor to object I1

Applying routine factor to object I2

Applying routine factor to object I3

Applying routine factor to object B(up,up)

Applying routine factor to object stress(up,up)

> grdisplay(_);

Because of the form of the invariants $I$,$\Phi$, $\psi$ and $p$ are constants. $p$ is solved from $\tau^{33} = 0$:

> P:=solve(grcomponent(stress(up,up),[3,3])=0,p):

> grmap(stress(up,up),subs,p=P,`x`):

Applying routine subs to stress(up,up)

> grdisplay(stress(up,up)):

The normal component of applied force on the side of the cube is given by $\tau^{11}/G^{11}$.

> grcomponent(stress(up,up),[1,1])/grcomponent(g(up,up)[deformed],[1,1]);

The tangential component is evaluated from $\tau^{1a}G_{a2}/ \surd{G_{11}G_{22}}$. This is calculated as follows:

> grdef(`L{^a c}:=stress{^a ^b}*g{b c}`):

Created definition for L(up,dn)

> grcalc(L(up,dn)):

> grcomponent(L(up,dn),[1,2])/(grcomponent(g(up,up),[1,1])^(1/2)*grcomponent(g(dn,dn),[2,2])^(1/2));

>