# Nonlinear finite elements/Stress and strain in one and two dimension
## 1-D strain measures

* Engineering strain :
    $\varepsilon _{E}:={\cfrac {l-L}{L}}={\cfrac {\Delta L}{L}}$
* Natural/Logarithmic/True strain:
    $\varepsilon _{L}:=\int _{L}^{l}{\cfrac {dl}{l}}=\ln(l){\Big |}_{L}^{l}=\ln \left({\cfrac {l}{L}}\right)$
Relation between engineering and true strain:
    $\varepsilon _{L}=\ln \left({\cfrac {L+\Delta L}{L}}\right)=\ln \left(1+{\cfrac {\Delta L}{L}}\right)=\ln \left(1+\varepsilon _{E}\right)$
* Green (Lagrangian) strain :
    $\varepsilon _{G}:={\frac {1}{2}}\left({\cfrac {l^{2}-L^{2}}{L^{2}}}\right)={\frac {1}{2}}\left({\cfrac {l^{2}}{L^{2}}}-1\right)$
* Almansi-Hamel (Eulerian) strain :
    $\varepsilon _{A}:={\frac {1}{2}}\left({\cfrac {l^{2}-L^{2}}{l^{2}}}\right)={\frac {1}{2}}\left(1-{\cfrac {L^{2}}{l^{2}}}\right)$

## 1-D stress measures

* Engineering/Nominal stress:
    $P=\sigma _{E}:={\cfrac {T}{A}}$
* Cauchy/True stress:
    $\sigma =\sigma _{T}:={\cfrac {T}{a}}$
Relation between engineering and true stress (no volume change):
    $\sigma _{T}={\cfrac {T}{a}}={\cfrac {Tl}{AL}}=\sigma _{E}\left({\cfrac {L+\Delta L}{L}}\right)=\sigma _{E}\left(1+\varepsilon _{E}\right)$

## 1-D stress-strain relations

* True stress - Green strain:
    $\sigma _{T}=E_{TG}\left({\cfrac {l^{2}-L^{2}}{2L^{2}}}\right)$
* True stress - True strain:
    $\sigma _{T}=E_{TT}\ln \left({\cfrac {l}{L}}\right)$

## Example

* Stretching of a bar
    $cock$
    $\sin \theta ={\cfrac {x}{l(x)}}$
* Assume incompressible material.
    $V=v~;~~AL=al$or :$V=al~;~~a=V/l$}
* Equilibrium.
    $T(x)=F$:$\implies ~~R(x)=T(x)-F=0$
where
    ${\begin{aligned}T(x)&=\sigma (x)~a(x)~\sin \theta \\&={\cfrac {\sigma (x)~a(x)~x}{l(x)}}\\&={\cfrac {\sigma (x)~V~x}{l(x)^{2}}}\end{aligned}}$

## Stress-strain relation 1

    $\sigma (x)=E\left({\cfrac {l(x)^{2}-L^{2}}{2L^{2}}}\right)$
Then,
    ${\begin{aligned}T(x)&={\cfrac {E~a(x)~x}{l(x)}}\left({\cfrac {l(x)^{2}-L^{2}}{2L^{2}}}\right)\\&={\cfrac {E~V~x}{l^{2}}}\left({\cfrac {l^{2}-L^{2}}{2L^{2}}}\right)\end{aligned}}$
and
    $R(x)={\cfrac {E~V~x}{l^{2}}}\left({\cfrac {l^{2}-L^{2}}{2L^{2}}}\right)-F$
Highly nonlinear in $x$.

## Stiffness

Stiffness = change in equilibrium equation due to change in position.
    $K(x)={\cfrac {dR(x)}{dx}}={\cfrac {dT(x)}{dx}}~~({\text{if}}~F~{\text{ is constant}})$
Now,
    $T(x)={\cfrac {\sigma (x)~V~x}{l(x)^{2}}}$
Therefore,
    ${\begin{aligned}{\cfrac {dT}{dx}}&=V\left[x~{\cfrac {d}{dx}}\left({\cfrac {\sigma }{l^{2}}}\right)+{\cfrac {\sigma }{l^{2}}}\right]\\&=V\left[x~{\cfrac {d}{d\sigma }}\left({\cfrac {\sigma }{l^{2}}}\right){\cfrac {d\sigma }{dx}}+x~{\cfrac {d}{dl}}\left({\cfrac {\sigma }{l^{2}}}\right){\cfrac {dl}{dx}}+{\cfrac {\sigma }{l^{2}}}\right]=V\left[{\cfrac {x}{l^{2}}}{\cfrac {d\sigma }{dl}}{\cfrac {dl}{dx}}-{\cfrac {2x\sigma }{l^{3}}}{\cfrac {dl}{dx}}+{\cfrac {\sigma }{l^{2}}}\right]\\&={\cfrac {Vx}{l^{2}}}\left({\cfrac {d\sigma }{dl}}-{\cfrac {2\sigma }{l}}\right){\cfrac {dl}{dx}}+{\cfrac {V\sigma }{l^{2}}}={\cfrac {Vx}{l^{2}}}\left({\cfrac {d\sigma }{dl}}-{\cfrac {2\sigma }{l}}\right){\cfrac {x}{l}}+{\cfrac {V\sigma }{l^{2}}}\\\implies K&=a\left({\cfrac {d\sigma }{dl}}-{\cfrac {2\sigma }{l}}\right){\cfrac {x^{2}}{l^{2}}}+{\cfrac {\sigma ~a}{l}}\end{aligned}}$
    $\sigma =E\left({\cfrac {l^{2}-L^{2}}{2L^{2}}}\right)\implies {\cfrac {d\sigma }{dl}}={\cfrac {E~l}{L^{2}}}$
    ${\begin{aligned}K&=a\left({\cfrac {d\sigma }{dl}}-{\cfrac {2\sigma }{l}}\right){\cfrac {x^{2}}{l^{2}}}+{\cfrac {\sigma ~a}{l}}\\&=a\left({\cfrac {E~l}{L^{2}}}-{\cfrac {2\sigma }{l}}\right){\cfrac {x^{2}}{l^{2}}}+{\cfrac {\sigma ~a}{l}}\\&={\cfrac {A}{L}}\left(E-2S\right){\cfrac {x^{2}}{l^{2}}}+{\cfrac {S~A}{L}}\end{aligned}}$
Initial stress/Geometric stiffness
    ${\cfrac {S~A}{L}}~;~~S=\sigma {\cfrac {L^{2}}{l^{2}}}=P{\cfrac {L}{l}}$

## Stress-strain relation 2

    $\sigma (x)=E\ln \left({\cfrac {l(x)}{L}}\right)$
Then,
    ${\begin{aligned}T(x)&={\cfrac {E~a(x)~x}{l(x)}}\ln \left({\cfrac {l(x)}{L}}\right)\\&={\cfrac {E~V~x}{l^{2}}}\ln \left({\cfrac {l}{L}}\right)\end{aligned}}$
and
    $R(x)={\cfrac {E~V~x}{l^{2}}}\ln \left({\cfrac {l}{L}}\right)-F$
Highly nonlinear in $x$.
    $\sigma =E\ln \left({\cfrac {l}{L}}\right)\implies {\cfrac {d\sigma }{dl}}={\cfrac {E}{l}}$

## Stiffness

    ${\begin{aligned}K&=a\left({\cfrac {d\sigma }{dl}}-{\cfrac {2\sigma }{l}}\right){\cfrac {x^{2}}{l^{2}}}+{\cfrac {\sigma ~a}{l}}\\&=a\left({\cfrac {E}{l}}-{\cfrac {2\sigma }{l}}\right){\cfrac {x^{2}}{l^{2}}}+{\cfrac {\sigma ~a}{l}}\\&={\cfrac {a}{l}}\left(E-2\sigma \right){\cfrac {x^{2}}{l^{2}}}+{\cfrac {\sigma ~a}{l}}\end{aligned}}$
Initial stress/Geometric stiffness:
    ${\cfrac {\sigma ~a}{l}}$

## Strain measures in two dimensions

* Strains in two dimensions

## Small strains

    ${\begin{aligned}\varepsilon _{xx}&={\frac {\partial u_{x}}{\partial x}}\\\varepsilon _{yy}&={\frac {\partial u_{y}}{\partial y}}\\\varepsilon _{xy}&={\frac {1}{2}}\left({\frac {\partial u_{x}}{\partial y}}+{\frac {\partial u_{y}}{\partial x}}\right)\end{aligned}}$
For 90$^{o}$ rotation,
    $u_{x}=-Y-X~;~~u_{y}=X-Y$
Then strains are:
    ${\begin{aligned}\varepsilon _{xx}&=-1\\\varepsilon _{yy}&=-1\\\varepsilon _{xy}&=0\end{aligned}}$
Rotation should not lead to non-zero strains!

## Finite strains

* Green strain in two dimensions
For 90$^{o}$ rotation,
    $u_{x}=-Y-X~;~~u_{y}=X-Y$
Then,
    $E_{xx}=0~;~~E_{yy}=0~;~~E_{xy}=0$

## Green strain (1-D)

    $\varepsilon _{G}={\cfrac {l^{2}-L^{2}}{2L^{2}}}$
In 2-D:
    $E_{xx}={\cfrac {ds^{2}-dX^{2}}{2dX^{2}}}$
Now,
    $ds^{2}=dX^{2}\left(1+{\frac {\partial u_{x}}{\partial X}}\right)^{2}+dX^{2}\left({\frac {\partial u_{y}}{\partial X}}\right)^{2}$
Therefore,
    ${\begin{aligned}E_{xx}&={\cfrac {dX^{2}}{2dX^{2}}}\left[\left(1+{\frac {\partial u_{x}}{\partial X}}\right)^{2}+\left({\frac {\partial u_{y}}{\partial X}}\right)^{2}-1\right]\\&={\frac {\partial u_{x}}{\partial X}}+{\frac {1}{2}}\left[\left({\frac {\partial u_{x}}{\partial X}}\right)^{2}+\left({\frac {\partial u_{y}}{\partial X}}\right)^{2}\right]\end{aligned}}$
Similar for $E_{yy}$ and $E_{xy}$.