# Nonlinear finite elements/Homework 10/Solutions
## Problem 1: Kinematics and Stress Rates

Given:
Figure 1 shows a linear three-noded triangular element in the reference configuration.
* Figure 1. Three-noded triangular element.
The motion of the nodes is given by:
    ${\begin{aligned}{\text{Node}}~1:&&x_{1}(t)&=0&\qquad y_{1}(t)&=0\\{\text{Node}}~2:&&x_{2}(t)&=2(1+at)\cos {\cfrac {\pi t}{2}}&\qquad y_{2}(t)&=2(1+at)\sin {\cfrac {\pi t}{2}}\\{\text{Node}}~3:&&x_{3}(t)&=-(1+bt)\sin {\cfrac {\pi t}{2}}&\qquad y_{3}(t)&=(1+bt)\cos {\cfrac {\pi t}{2}}\end{aligned}}$
The configuration ($x,y$) of the element at time $t$ is given by
    $x({\boldsymbol {\xi }},t)=\sum _{i=1}^{3}x_{i}(t)~N_{i}({\boldsymbol {\xi }})~;\qquad y({\boldsymbol {\xi }},t)=\sum _{i=1}^{3}y_{i}(t)~N_{i}({\boldsymbol {\xi }})$

### Solutions
#### Part 1

Write down expressions for $N_{1}$, $N_{2}$, and $N_{3}$ in terms of the initial configuration ($X,Y$) ?
In the initial configuration $t=0$. Therefore,
    $X_{1}=x_{1}(0)=0~;~~Y_{1}=y_{1}(0)=0~;\qquad X_{2}=x_{2}(0)=2~;~~Y_{2}=y_{2}(0)=0~;\qquad X_{3}=x_{3}(0)=0~;~~Y_{3}=y_{3}(0)=1~.$
Therefore, the initial configuration is given by
    $X({\boldsymbol {\xi }})=\sum _{i=1}^{3}X_{i}~N_{i}({\boldsymbol {\xi }})~;\qquad Y({\boldsymbol {\xi }})=\sum _{i=1}^{3}Y_{i}~N_{i}({\boldsymbol {\xi }})~.$
Substituting the values of $X_{i}$ and $Y_{i}$, we get
    $X({\boldsymbol {\xi }})=2~N_{2}({\boldsymbol {\xi }})~;\qquad Y({\boldsymbol {\xi }})=N_{3}({\boldsymbol {\xi }})~.$
Hence
    $N_{2}({\boldsymbol {\xi }})={\cfrac {X({\boldsymbol {\xi }})}{2}}~;\qquad N_{3}({\boldsymbol {\xi }})=Y({\boldsymbol {\xi }})~.$
Also, the shape functions must satisfy the partition of unity condition
    $\sum _{i=1}^{3}N_{i}({\boldsymbol {\xi }})=1~.$
Therefore,
    $N_{1}({\boldsymbol {\xi }})=1-N_{2}({\boldsymbol {\xi }})-N_{3}({\boldsymbol {\xi }})=1-{\cfrac {X({\boldsymbol {\xi }})}{2}}-Y({\boldsymbol {\xi }})~.$
The required expressions are
    ${N_{1}({\boldsymbol {\xi }})=1-{\cfrac {X({\boldsymbol {\xi }})}{2}}-Y({\boldsymbol {\xi }})~;\qquad N_{2}({\boldsymbol {\xi }})={\cfrac {X({\boldsymbol {\xi }})}{2}}~;\qquad N_{3}({\boldsymbol {\xi }})=Y({\boldsymbol {\xi }})~.}$

#### Part 2

Derive expressions for the deformation gradient and the Jacobian determinant for the element as functions of time.
The deformation gradient is given by
    $\mathbf {F} =\left[{\begin{aligned}{\frac {\partial x}{\partial X}}&&{\frac {\partial x}{\partial Y}}&&{\frac {\partial x}{\partial Z}}\\{\frac {\partial y}{\partial X}}&&{\frac {\partial y}{\partial Y}}&&{\frac {\partial y}{\partial Z}}\\{\frac {\partial z}{\partial X}}&&{\frac {\partial z}{\partial Y}}&&{\frac {\partial z}{\partial Z}}\end{aligned}}\right]~.$
Before computing the derivatives, let us express $x,y,z$ in terms of $X,Y,Z$. Recall
    $x({\boldsymbol {\xi }},t)=\sum _{i=1}^{3}x_{i}(t)~N_{i}({\boldsymbol {\xi }})~;\qquad y({\boldsymbol {\xi }},t)=\sum _{i=1}^{3}y_{i}(t)~N_{i}({\boldsymbol {\xi }})$
Therefore,
    ${\begin{aligned}x({\boldsymbol {\xi }},t)&=x_{1}(t)~\left[1-{\cfrac {X({\boldsymbol {\xi }})}{2}}-Y({\boldsymbol {\xi }})\right]+x_{2}(t)~{\cfrac {X({\boldsymbol {\xi }})}{2}}+x_{3}(t)~Y({\boldsymbol {\xi }})\\y({\boldsymbol {\xi }},t)&=y_{1}(t)~\left[1-{\cfrac {X({\boldsymbol {\xi }})}{2}}-Y({\boldsymbol {\xi }})\right]+y_{2}(t)~{\cfrac {X({\boldsymbol {\xi }})}{2}}+y_{3}(t)~Y({\boldsymbol {\xi }})\\z({\boldsymbol {\xi }},t)&=Z({\boldsymbol {\xi }})\end{aligned}}$
Substituting in the expressions for $x_{i}$ and $y_{i}$, we get
    ${\begin{aligned}x({\boldsymbol {\xi }},t)&=\left[2(1+at)\cos {\cfrac {\pi t}{2}}\right]~{\cfrac {X({\boldsymbol {\xi }})}{2}}-\left[(1+bt)\sin {\cfrac {\pi t}{2}}\right]~Y({\boldsymbol {\xi }})\\y({\boldsymbol {\xi }},t)&=\left[2(1+at)\sin {\cfrac {\pi t}{2}}\right]~{\cfrac {X({\boldsymbol {\xi }})}{2}}+\left[(1+bt)\cos {\cfrac {\pi t}{2}}\right]~Y({\boldsymbol {\xi }})\\z({\boldsymbol {\xi }},t)&=Z({\boldsymbol {\xi }})~.\end{aligned}}$
In the above expression, the parent coordinates ${\boldsymbol {\xi }}$ are no longer useful. Therefore, we write
    ${\begin{aligned}x(\mathbf {X} ,t)&=X~(1+at)~\cos {\cfrac {\pi t}{2}}-Y~(1+bt)~\sin {\cfrac {\pi t}{2}}\\y(\mathbf {X} ,t)&=X~(1+at)~\sin {\cfrac {\pi t}{2}}+Y~(1+bt)~\cos {\cfrac {\pi t}{2}}\\z(\mathbf {X} ,t)&=Z\end{aligned}}$
where $\mathbf {X} =(X,Y,Z)$. Taking derivatives, we get
    ${\begin{aligned}{\frac {\partial x}{\partial X}}&=(1+at)~\cos {\cfrac {\pi t}{2}}\qquad &{\frac {\partial x}{\partial Y}}&=-(1+bt)~\sin {\cfrac {\pi t}{2}}\qquad &{\frac {\partial x}{\partial Z}}&=0\\{\frac {\partial y}{\partial X}}&=(1+at)~\sin {\cfrac {\pi t}{2}}\qquad &{\frac {\partial y}{\partial Y}}&=(1+bt)~\cos {\cfrac {\pi t}{2}}\qquad &{\frac {\partial y}{\partial Z}}&=0\\{\frac {\partial z}{\partial X}}&=0\qquad &{\frac {\partial z}{\partial Y}}&=0\qquad &{\frac {\partial z}{\partial Z}}&=1\end{aligned}}$
Therefore,
    ${\mathbf {F} (t)=\left[{\begin{aligned}(1+at)~\cos {\cfrac {\pi t}{2}}&&-(1+bt)~\sin {\cfrac {\pi t}{2}}&&0\\(1+at)~\sin {\cfrac {\pi t}{2}}&&(1+bt)~\cos {\cfrac {\pi t}{2}}&&0\\0&&0&&1\end{aligned}}\right]~.}$
The Jacobian determinant is
    $J(t)=\det \mathbf {F} (t)=\left[(1+at)~\cos {\cfrac {\pi t}{2}}\right]\left[(1+bt)~\cos {\cfrac {\pi t}{2}}\right]+\left[(1+bt)~\sin {\cfrac {\pi t}{2}}\right]\left[(1+at)~\sin {\cfrac {\pi t}{2}}\right]~.$
or
    ${J(t)=(1+at)(1+bt)~.}$

#### Part 3

What are the values of $a$ and $b$ for which the motion is isochoric?
For isochoric motion, $J=1$. Therefore,
    $(1+at)(1+bt)=1~.$
One possibility is
    ${a=b=0}$
This is a pure rotation.
Another possibility is that
    ${b=-{\cfrac {a}{1+at}}}$
This is a combination of shear and rotation where the volume remains constant.

#### Part 4

For which values of $a$ and $b$ do we get invalid motions?
We get invalid motions when $J\leq 0$. Let us consider the case where $J=0$. Then
    $(1+at)(1+bt)=0$
This is possible when
    $at=-1~{\text{or}}~bt=-1~.$
If $J<0$ then
    $1+at<0~{\text{and}}~1+bt>0\qquad {\text{or}}\qquad 1+at>0~{\text{and}}~1+bt<0$
That is,
    $at<-1~{\text{and}}~bt>-1\qquad {\text{or}}\qquad at>-1~{\text{and}}~bt<-1$
Therefore the values at which we get invalid motions are
    ${a\leq -{\cfrac {1}{t}}~~{\text{or}}~~b\leq -{\cfrac {1}{t}}~.}$

#### Part 5

for the element as a function of time.
The Green strain tensor is given by
    $\mathbf {E} ={\frac {1}{2}}(\mathbf {F} ^{T}\mathbf {F} -\mathbf {I} )$
Recall
    $\mathbf {F} (t)=\left[{\begin{aligned}(1+at)~\cos {\cfrac {\pi t}{2}}&&-(1+bt)~\sin {\cfrac {\pi t}{2}}&&0\\(1+at)~\sin {\cfrac {\pi t}{2}}&&(1+bt)~\cos {\cfrac {\pi t}{2}}&&0\\0&&0&&1\end{aligned}}\right]~.$
Let us make the following substitutions
    $A:=(1+at)~;~~B:=(1+bt)~;~~c:=\cos {\cfrac {\pi t}{2}}~;~~s:=\sin {\cfrac {\pi t}{2}}~.$
Then
    $\mathbf {F} ={\begin{bmatrix}A~c&-B~s&0\\A~s&B~c&0\\0&0&1\end{bmatrix}}~~{\text{and}}~~\mathbf {F} ^{T}={\begin{bmatrix}A~c&A~s&0\\-B~s&B~c&0\\0&0&1\end{bmatrix}}~.$
Therefore,
    $\mathbf {F} ^{T}~\mathbf {F} ={\begin{bmatrix}A^{2}&0&0\\0&B^{2}&0\\0&0&1\end{bmatrix}}={\begin{bmatrix}(1+at)^{2}&0&0\\0&(1+bt)^{2}&0\\0&0&1\end{bmatrix}}~.$
Hence,
    $\mathbf {E} ={\frac {1}{2}}\left({\begin{bmatrix}(1+at)^{2}&0&0\\0&(1+bt)^{2}&0\\0&0&1\end{bmatrix}}-{\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}}\right)$
The Green strain is
    ${\mathbf {E} (t)={\frac {1}{2}}{\begin{bmatrix}2at+a^{2}t^{2}&0&0\\0&2bt+b^{2}t^{2}&0\\0&0&0\end{bmatrix}}}$

#### Part 6

time.
The velocity is the material time derivative of the motion.
Recall that the motion is given by
    ${\begin{aligned}x(\mathbf {X} ,t)&=X~(1+at)~\cos {\cfrac {\pi t}{2}}-Y~(1+bt)~\sin {\cfrac {\pi t}{2}}\\y(\mathbf {X} ,t)&=X~(1+at)~\sin {\cfrac {\pi t}{2}}+Y~(1+bt)~\cos {\cfrac {\pi t}{2}}\\z(\mathbf {X} ,t)&=Z\end{aligned}}$
Therefore,
    ${\begin{aligned}v_{x}(\mathbf {X} ,t)={\frac {\partial x}{\partial t}}&=X~\left[a~\cos {\cfrac {\pi t}{2}}-(1+at)~{\cfrac {\pi }{2}}~\sin {\cfrac {\pi t}{2}}\right]-Y~\left[b~\sin {\cfrac {\pi t}{2}}+(1+bt)~{\cfrac {\pi }{2}}~\cos {\cfrac {\pi t}{2}}\right]\\v_{y}(\mathbf {X} ,t)={\frac {\partial y}{\partial t}}&=X~\left[a~\sin {\cfrac {\pi t}{2}}+(1+at)~{\cfrac {\pi }{2}}~\cos {\cfrac {\pi t}{2}}\right]+Y~\left[b~\cos {\cfrac {\pi t}{2}}-(1+bt)~{\cfrac {\pi }{2}}~\sin {\cfrac {\pi t}{2}}\right]\\v_{z}\mathbf {X} ,t)={\frac {\partial z}{\partial t}}&=0\end{aligned}}$
We could compute the velocity gradient using
    $\mathbf {l} ={\boldsymbol {\nabla }}\mathbf {v} =\left[{\begin{aligned}{\frac {\partial v_{x}}{\partial x}}&&{\frac {\partial v_{x}}{\partial y}}&&{\frac {\partial v_{x}}{\partial z}}\\{\frac {\partial v_{y}}{\partial x}}&&{\frac {\partial v_{y}}{\partial y}}&&{\frac {\partial v_{y}}{\partial z}}\\{\frac {\partial v_{z}}{\partial x}}&&{\frac {\partial v_{z}}{\partial y}}&&{\frac {\partial v_{z}}{\partial z}}\end{aligned}}\right]$
after expressing $\mathbf {v}$ in terms of $\mathbf {x}$. However, that makes the expression quite complicated. Instead, we will use the relation
    $\mathbf {l} ={\dot {\mathbf {F} }}~\mathbf {F} ^{-1}~.$
The time derivative of the deformation gradient is
    ${\dot {\mathbf {F} }}=\left[{\begin{aligned}a~\cos {\cfrac {\pi t}{2}}-(1+at)~{\cfrac {\pi }{2}}~\sin {\cfrac {\pi t}{2}}&&-b~\sin {\cfrac {\pi t}{2}}-(1+bt)~{\cfrac {\pi }{2}}~\cos {\cfrac {\pi t}{2}}&&0\\a~\sin {\cfrac {\pi t}{2}}+(1+at)~{\cfrac {\pi }{2}}~\cos {\cfrac {\pi t}{2}}&&b~\cos {\cfrac {\pi t}{2}}-(1+bt)~{\cfrac {\pi }{2}}~\sin {\cfrac {\pi t}{2}}&&0\\0&&0&&0\end{aligned}}\right]~.$
The inverse of the deformation gradient is
    $\mathbf {F} ^{-1}={\cfrac {1}{(1+at)(1+bt)}}\left[{\begin{aligned}(1+bt)~\cos {\cfrac {\pi t}{2}}&&(1+bt)~\sin {\cfrac {\pi t}{2}}&&0\\-(1+at)~\sin {\cfrac {\pi t}{2}}&&(1+at)~\cos {\cfrac {\pi t}{2}}&&0\\0&&0&&(1+at)(1+bt)\end{aligned}}\right]~.$
Using the substitutions
    $A:=(1+at)~;~~B:=(1+bt)~;~~c:=\cos {\cfrac {\pi t}{2}}~;~~s:=\sin {\cfrac {\pi t}{2}}$
we get
    ${\dot {\mathbf {F} }}=\left[{\begin{aligned}a~c-A~s~{\cfrac {\pi }{2}}&&-b~s-B~c~{\cfrac {\pi }{2}}&&0\\a~s+A~c~{\cfrac {\pi }{2}}&&b~c-B~s~{\cfrac {\pi }{2}}&&0\\0&&0&&0\end{aligned}}\right]=\left[{\begin{aligned}a~c&&-b~s&&0\\a~s&&b~c&&0\\0&&0&&0\end{aligned}}\right]+{\cfrac {\pi }{2}}\left[{\begin{aligned}-A~s&&-B~c&&0\\A~c&&-B~s&&0\\0&&0&&0\end{aligned}}\right]$
and
    $\mathbf {F} ^{-1}={\cfrac {1}{A~B}}\left[{\begin{aligned}B~c&&B~s&&0\\-A~s&&A~c&&0\\0&&0&&A~B\end{aligned}}\right]~.$
The product is
    $\mathbf {l} ={\cfrac {1}{A~B}}\left[{\begin{aligned}B~a~c^{2}+A~b~s^{2}&&B~a~c~s-A~b~c~s&&0\\B~a~c~s-A~b~c~s&&B~a~s^{2}+A~b~c^{2}&&0\\0&&0&&0\end{aligned}}\right]+{\cfrac {\pi }{2}}\left[{\begin{aligned}0&&-1&&0\\1&&0&&0\\0&&0&&0\end{aligned}}\right]$
Note that the first matrix is symmetric while the second is skew-symmetric.
Therefore, the velocity gradient is
    ${\mathbf {l} =\left[{\begin{aligned}{\cfrac {a}{1+at}}~\cos ^{2}{\cfrac {\pi t}{2}}+{\cfrac {b}{1+bt}}~\sin ^{2}{\cfrac {\pi t}{2}}&&\left[{\cfrac {a}{1+at}}-{\cfrac {b}{1+bt}}\right]\cos {\cfrac {\pi t}{2}}~\sin {\cfrac {\pi t}{2}}-{\frac {\pi }{2}}&&0\\\left[{\cfrac {a}{1+at}}-{\cfrac {b}{1+bt}}\right]\cos {\cfrac {\pi t}{2}}~\sin {\cfrac {\pi t}{2}}+{\cfrac {\pi }{2}}&&{\cfrac {a}{1+at}}~\sin ^{2}{\cfrac {\pi t}{2}}+{\cfrac {b}{1+bt}}~\cos ^{2}{\cfrac {\pi t}{2}}&&0\\0&&0&&0\end{aligned}}\right]}$

#### Part 7

Compute the rate of deformation tensor and the spin tensor.
The rate of deformation is the symmetric part of the velocity gradient:
    ${\mathbf {d} =\left[{\begin{aligned}{\cfrac {a}{1+at}}~\cos ^{2}{\cfrac {\pi t}{2}}+{\cfrac {b}{1+bt}}~\sin ^{2}{\cfrac {\pi t}{2}}&&\left[{\cfrac {a}{1+at}}-{\cfrac {b}{1+bt}}\right]\cos {\cfrac {\pi t}{2}}~\sin {\cfrac {\pi t}{2}}&&0\\\left[{\cfrac {a}{1+at}}-{\cfrac {b}{1+bt}}\right]\cos {\cfrac {\pi t}{2}}~\sin {\cfrac {\pi t}{2}}&&{\cfrac {a}{1+at}}~\sin ^{2}{\cfrac {\pi t}{2}}+{\cfrac {b}{1+bt}}~\cos ^{2}{\cfrac {\pi t}{2}}&&0\\0&&0&&0\end{aligned}}\right]}$
The rate of deformation is the skew-symmetric part of the velocity gradient:
    ${\mathbf {w} ={\cfrac {\pi }{2}}\left[{\begin{aligned}0&&-1&&0\\1&&0&&0\\0&&0&&0\end{aligned}}\right]}$

#### Part 8

and label.
Recall that in the initial configuration
    $X_{1}=x_{1}(0)=0~;~~Y_{1}=y_{1}(0)=0~;\qquad X_{2}=x_{2}(0)=2~;~~Y_{2}=y_{2}(0)=0~;\qquad X_{3}=x_{3}(0)=0~;~~Y_{3}=y_{3}(0)=1~.$
Also, the motion is
    ${\begin{aligned}x(\mathbf {X} ,t)&=X~(1+at)~\cos {\cfrac {\pi t}{2}}-Y~(1+bt)~\sin {\cfrac {\pi t}{2}}\\y(\mathbf {X} ,t)&=X~(1+at)~\sin {\cfrac {\pi t}{2}}+Y~(1+bt)~\cos {\cfrac {\pi t}{2}}\\z(\mathbf {X} ,t)&=Z\end{aligned}}$
Plugging in the values of $a$ and $b$, we get
    ${\begin{aligned}x(\mathbf {X} ,t)&=X~(1+t)~\cos {\cfrac {\pi t}{2}}-Y~(1+2t)~\sin {\cfrac {\pi t}{2}}\\y(\mathbf {X} ,t)&=X~(1+t)~\sin {\cfrac {\pi t}{2}}+Y~(1+2t)~\cos {\cfrac {\pi t}{2}}\\z(\mathbf {X} ,t)&=Z\end{aligned}}$
At $t=1$,
    ${\begin{aligned}x(\mathbf {X} ,1)&=2X~\cos {\cfrac {\pi }{2}}-3Y~\sin {\cfrac {\pi }{2}}=-3Y\\y(\mathbf {X} ,1)&=2X~\sin {\cfrac {\pi }{2}}+3Y~\cos {\cfrac {\pi }{2}}=2X\\z(\mathbf {X} ,1)&=Z\end{aligned}}$
At $t=2$,
    ${\begin{aligned}x(\mathbf {X} ,2)&=3X~\cos \pi -5Y~\sin \pi =-3X\\y(\mathbf {X} ,2)&=3X~\sin \pi +5Y~\cos \pi =-5Y\\z(\mathbf {X} ,2)&=Z\end{aligned}}$
The deformed and undeformed configurations are shown below.
* Deformed and undeformed configurations.

#### Part 9

Compute the polar decomposition of the deformation gradient with the above values of $a$ and $b$,
The deformation gradient is
    $\mathbf {F} (t)=\left[{\begin{aligned}(1+t)~\cos {\cfrac {\pi t}{2}}&&-(1+2t)~\sin {\cfrac {\pi t}{2}}&&0\\(1+t)~\sin {\cfrac {\pi t}{2}}&&(1+2t)~\cos {\cfrac {\pi t}{2}}&&0\\0&&0&&1\end{aligned}}\right]~.$
The right Cauchy-Green deformation tensor is
    $\mathbf {C} =\mathbf {F} ^{T}~\mathbf {F} ={\begin{bmatrix}(1+t)^{2}&0&0\\0&(1+2t)^{2}&0\\0&0&1\end{bmatrix}}$
The eigenvalue problem is
    $(\mathbf {C} -\lambda ^{2}\mathbf {I} )\mathbf {N} =0~.$
This problem has a solution if
    $\det(\mathbf {C} -\lambda ^{2}\mathbf {I} )=0$
i.e.,
    $\left[\lambda ^{4}-(1+t)^{2}\lambda ^{2}-\lambda ^{2}(1+2t)^{2}+(1+t)^{2}~(1+2t)^{2}\right](1-\lambda ^{2})=0~.$
The eigenvalues are (as expected)
    $\lambda _{1}^{2}=(1+t)^{2}~;~~\lambda _{2}^{2}=(1+2t)^{2}~;~~\lambda _{3}^{2}=1~.$
The principal stretches are
    $\lambda _{1}=(1+t)~;~~\lambda _{2}=(1+2t)~;~~\lambda _{3}=1~.$
The principal directions are (by inspection)
    $\mathbf {N} _{1}=[1~~0~~0]^{T}~;~~\mathbf {N} _{2}=[0~~1~~0]^{T}~;~~\mathbf {N} _{3}=[0~~0~~1]^{T}~.$
Now, the right stretch tensor is given by
    ${\boldsymbol {U}}=\sum _{i=1}^{3}\lambda _{i}{\boldsymbol {N}}_{i}\otimes {\boldsymbol {N}}_{i}$
Therefore,
    $\mathbf {U} =\lambda _{1}~\mathbf {N} _{1}~\mathbf {N} _{1}^{T}+\lambda _{2}~\mathbf {N} _{2}~\mathbf {N} _{2}^{T}+\lambda _{3}~\mathbf {N} _{3}~\mathbf {N} _{3}^{T}~.$
Hence the right stretch is
    ${\mathbf {U} ={\begin{bmatrix}1+t&0&0\\0&1+2t&0\\0&0&1\end{bmatrix}}}$
At $t=0,1,2$, we have
    $\mathbf {U} (0)={\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}}~;~~\mathbf {U} (1)={\begin{bmatrix}2&0&0\\0&3&0\\0&0&1\end{bmatrix}}~;~~\mathbf {U} (2)={\begin{bmatrix}3&0&0\\0&5&0\\0&0&1\end{bmatrix}}~.$
Now
    $\mathbf {R} =\mathbf {F} ~\mathbf {U} ^{-1}$
and
    $\mathbf {U} ^{-1}={\begin{bmatrix}{\cfrac {1}{1+t}}&0&0\\0&{\cfrac {1}{1+2t}}&0\\0&0&1\end{bmatrix}}$
Therefore, the rotation is
    ${\mathbf {R} =\left[{\begin{aligned}\cos {\cfrac {\pi t}{2}}&&-\sin {\cfrac {\pi t}{2}}&&0\\\sin {\cfrac {\pi t}{2}}&&\cos {\cfrac {\pi t}{2}}&&0\\0&&0&&1\end{aligned}}\right]~.}$
At $t=0,1,2$, we have
    $\mathbf {R} (0)={\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}}~;~~\mathbf {R} (1)={\begin{bmatrix}0&-1&0\\1&0&0\\0&0&1\end{bmatrix}}~;~~\mathbf {R} (2)={\begin{bmatrix}-1&0&0\\0&-1&0\\0&0&1\end{bmatrix}}~.$

#### Part 10

Assume an isotropic, hypoelastic constitutive equation for the material of the element. Compute the material time derivative of the Cauchy stress at $t=1$ using (a) the Jaumann rate and (b) the Truesdell rate.
A hypoelastic material behaves according to the relation
    ${\overset {\triangle }{\boldsymbol {\sigma }}}={\boldsymbol {\mathsf {C}}}:\mathbf {d} ~.$
For an isotropic material
    ${\boldsymbol {\mathsf {C}}}=\lambda ~{\boldsymbol {\mathit {1}}}\otimes {\boldsymbol {\mathit {1}}}+2\mu ~{\boldsymbol {I}}$
Therefore,
    ${\overset {\triangle }{\boldsymbol {\sigma }}}=\lambda ~{\boldsymbol {\mathit {1}}}\otimes {\boldsymbol {\mathit {1}}}:\mathbf {d} +2\mu ~{\boldsymbol {I}}:\mathbf {d} =\lambda ~{\text{tr}}(\mathbf {d} )~{\boldsymbol {\mathit {1}}}+2\mu ~\mathbf {d}$
Recall that
    $\mathbf {d} =\left[{\begin{aligned}{\cfrac {a}{1+at}}~\cos ^{2}{\cfrac {\pi t}{2}}+{\cfrac {b}{1+bt}}~\sin ^{2}{\cfrac {\pi t}{2}}&&\left[{\cfrac {a}{1+at}}-{\cfrac {b}{1+bt}}\right]\cos {\cfrac {\pi t}{2}}~\sin {\cfrac {\pi t}{2}}&&0\\\left[{\cfrac {a}{1+at}}-{\cfrac {b}{1+bt}}\right]\cos {\cfrac {\pi t}{2}}~\sin {\cfrac {\pi t}{2}}&&{\cfrac {a}{1+at}}~a~\sin ^{2}{\cfrac {\pi t}{2}}+{\cfrac {b}{1+bt}}~\cos ^{2}{\cfrac {\pi t}{2}}&&0\\0&&0&&0\end{aligned}}\right]$
Using the values of $a$ and $b$ from the previous part, at $t=1$,
    $\mathbf {d} =\left[{\begin{aligned}{\cfrac {1}{2}}~\cos ^{2}{\cfrac {\pi }{2}}+{\cfrac {2}{3}}~\sin ^{2}{\cfrac {\pi }{2}}&&\left[{\cfrac {1}{2}}-{\cfrac {2}{3}}\right]\cos {\cfrac {\pi }{2}}~\sin {\cfrac {\pi }{2}}&&0\\\left[{\cfrac {1}{2}}-{\cfrac {2}{3}}\right]\cos {\cfrac {\pi }{2}}~\sin {\cfrac {\pi }{2}}&&{\cfrac {1}{2}}~\sin ^{2}{\cfrac {\pi }{2}}+{\cfrac {2}{3}}~\cos ^{2}{\cfrac {\pi }{2}}&&0\\0&&0&&0\end{aligned}}\right]={\begin{bmatrix}{\cfrac {2}{3}}&0&0\\0&{\cfrac {1}{2}}&0\\0&0&0\end{bmatrix}}$
Therefore, the trace of the rate of defromation is
    ${\text{tr}}(\mathbf {d} )={\cfrac {7}{6}}$
Therefore,
    ${\overset {\triangle }{\boldsymbol {\sigma }}}={\cfrac {7\lambda }{6}}~{\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}}+2\mu ~{\begin{bmatrix}{\cfrac {2}{3}}&0&0\\0&{\cfrac {1}{2}}&0\\0&0&0\end{bmatrix}}={\begin{bmatrix}{\cfrac {7\lambda }{6}}+{\cfrac {4\mu }{3}}&0&0\\0&{\cfrac {7\lambda }{6}}+\mu &0\\0&0&{\cfrac {7\lambda }{6}}\end{bmatrix}}$
For the Jaumann rate
    ${\overset {\triangle }{\boldsymbol {\sigma }}}={\dot {\boldsymbol {\sigma }}}+{\boldsymbol {\sigma }}\bullet \mathbf {w} -\mathbf {w} \bullet {\boldsymbol {\sigma }}$
where the spin is
    $\mathbf {w} ={\cfrac {\pi }{2}}\left[{\begin{aligned}0&&-1&&0\\1&&0&&0\\0&&0&&0\end{aligned}}\right]$
Therefore,
    ${\dot {\boldsymbol {\sigma }}}={\begin{bmatrix}{\cfrac {7\lambda }{6}}+{\cfrac {4\mu }{3}}&0&0\\0&{\cfrac {7\lambda }{6}}+\mu &0\\0&0&{\cfrac {7\lambda }{6}}\end{bmatrix}}-{\cfrac {\pi }{2}}{\begin{bmatrix}\sigma _{xx}&\sigma _{xy}&\sigma _{xz}\\\sigma _{xy}&\sigma _{yy}&\sigma _{yz}\\\sigma _{xz}&\sigma _{yz}&\sigma _{zz}\end{bmatrix}}{\begin{bmatrix}0&&-1&&0\\1&&0&&0\\0&&0&&0\end{bmatrix}}+{\cfrac {\pi }{2}}{\begin{bmatrix}0&&-1&&0\\1&&0&&0\\0&&0&&0\end{bmatrix}}{\begin{bmatrix}\sigma _{xx}&\sigma _{xy}&\sigma _{xz}\\\sigma _{xy}&\sigma _{yy}&\sigma _{yz}\\\sigma _{xz}&\sigma _{yz}&\sigma _{zz}\end{bmatrix}}$
or,
    ${{\dot {\boldsymbol {\sigma }}}={\begin{bmatrix}{\cfrac {7\lambda }{6}}+{\cfrac {4\mu }{3}}&0&0\\0&{\cfrac {7\lambda }{6}}+\mu &0\\0&0&{\cfrac {7\lambda }{6}}\end{bmatrix}}+{\cfrac {\pi }{2}}{\begin{bmatrix}-\sigma _{xy}&\sigma _{xx}-\sigma _{yy}&-\sigma _{yz}\\\sigma _{xx}-\sigma _{yy}&2\sigma _{xy}&\sigma _{xz}\\-\sigma _{yz}&\sigma _{xz}&0\end{bmatrix}}}$
For the Truesdell rate
    ${\overset {\triangle }{\boldsymbol {\sigma }}}={\overset {\circ }{\boldsymbol {\sigma }}}={\dot {\boldsymbol {\sigma }}}-{\boldsymbol {\sigma }}\bullet \mathbf {l} ^{T}-\mathbf {l} \bullet {\boldsymbol {\sigma }}+{\text{tr}}(\mathbf {l} ){\boldsymbol {\sigma }}~.$
Therefore,
    ${\dot {\boldsymbol {\sigma }}}={\overset {\circ }{\boldsymbol {\sigma }}}+{\boldsymbol {\sigma }}\bullet \mathbf {l} ^{T}+\mathbf {l} \bullet {\boldsymbol {\sigma }}-{\text{tr}}(\mathbf {l} ){\boldsymbol {\sigma }}~.$
Recall,
    $\mathbf {l} =\left[{\begin{aligned}{\cfrac {a}{1+at}}~\cos ^{2}{\cfrac {\pi t}{2}}+{\cfrac {b}{1+bt}}~\sin ^{2}{\cfrac {\pi t}{2}}&&\left[{\cfrac {a}{1+at}}-{\cfrac {b}{1+bt}}\right]\cos {\cfrac {\pi t}{2}}~\sin {\cfrac {\pi t}{2}}-{\frac {\pi }{2}}&&0\\\left[{\cfrac {a}{1+at}}-{\cfrac {b}{1+bt}}\right]\cos {\cfrac {\pi t}{2}}~\sin {\cfrac {\pi t}{2}}+{\cfrac {\pi }{2}}&&{\cfrac {a}{1+at}}~\sin ^{2}{\cfrac {\pi t}{2}}+{\cfrac {b}{1+bt}}~\cos ^{2}{\cfrac {\pi t}{2}}&&0\\0&&0&&0\end{aligned}}\right]$
For $a=1$, $b=2$, $t=1$, we have
    $\mathbf {l} =\left[{\begin{aligned}{\cfrac {1}{2}}~\cos ^{2}{\cfrac {\pi }{2}}+{\cfrac {2}{3}}~\sin ^{2}{\cfrac {\pi }{2}}&&\left[{\cfrac {1}{2}}-{\cfrac {2}{3}}\right]\cos {\cfrac {\pi }{2}}~\sin {\cfrac {\pi }{2}}-{\frac {\pi }{2}}&&0\\\left[{\cfrac {1}{2}}-{\cfrac {2}{3}}\right]\cos {\cfrac {\pi }{2}}~\sin {\cfrac {\pi }{2}}+{\cfrac {\pi }{2}}&&{\cfrac {1}{2}}~\sin ^{2}{\cfrac {\pi }{2}}+{\cfrac {2}{3}}~\cos ^{2}{\cfrac {\pi }{2}}&&0\\0&&0&&0\end{aligned}}\right]={\begin{bmatrix}{\cfrac {2}{3}}&-{\cfrac {\pi }{2}}&0\\{\cfrac {\pi }{2}}&{\cfrac {1}{2}}&0\\0&0&0\end{bmatrix}}~.$
Therefore,
    ${\text{tr}}(\mathbf {l} )={\cfrac {7}{6}}={\text{tr}}(\mathbf {d} )$
and
    ${\dot {\boldsymbol {\sigma }}}={\begin{bmatrix}{\cfrac {7\lambda }{6}}+{\cfrac {4\mu }{3}}&0&0\\0&{\cfrac {7\lambda }{6}}+\mu &0\\0&0&{\cfrac {7\lambda }{6}}\end{bmatrix}}+{\begin{bmatrix}\sigma _{xx}&\sigma _{xy}&\sigma _{xz}\\\sigma _{xy}&\sigma _{yy}&\sigma _{yz}\\\sigma _{xz}&\sigma _{yz}&\sigma _{zz}\end{bmatrix}}{\begin{bmatrix}{\cfrac {2}{3}}&{\cfrac {\pi }{2}}&0\\-{\cfrac {\pi }{2}}&{\cfrac {1}{2}}&0\\0&0&0\end{bmatrix}}+{\begin{bmatrix}{\cfrac {2}{3}}&-{\cfrac {\pi }{2}}&0\\{\cfrac {\pi }{2}}&{\cfrac {1}{2}}&0\\0&0&0\end{bmatrix}}{\begin{bmatrix}\sigma _{xx}&\sigma _{xy}&\sigma _{xz}\\\sigma _{xy}&\sigma _{yy}&\sigma _{yz}\\\sigma _{xz}&\sigma _{yz}&\sigma _{zz}\end{bmatrix}}-{\cfrac {7}{6}}{\begin{bmatrix}\sigma _{xx}&\sigma _{xy}&\sigma _{xz}\\\sigma _{xy}&\sigma _{yy}&\sigma _{yz}\\\sigma _{xz}&\sigma _{yz}&\sigma _{zz}\end{bmatrix}}$
Hence,
    ${{\dot {\boldsymbol {\sigma }}}={\begin{bmatrix}{\cfrac {7\lambda }{6}}+{\cfrac {4\mu }{3}}&0&0\\0&{\cfrac {7\lambda }{6}}+\mu &0\\0&0&{\cfrac {7\lambda }{6}}\end{bmatrix}}+{\begin{bmatrix}{\cfrac {1}{6}}\sigma _{xx}+\pi \sigma _{xy}&{\cfrac {\pi }{2}}(-\sigma _{xx}+\sigma _{yy})&{\cfrac {1}{2}}(-\sigma _{xz}+\pi \sigma _{yz})\\{\cfrac {\pi }{2}}(-\sigma _{xx}+\sigma _{yy})&-{\cfrac {1}{6}}\sigma _{yy}-\pi \sigma _{xy}&-{\cfrac {2}{3}}\sigma _{yz}-{\cfrac {\pi }{2}}\sigma _{xz}\\{\cfrac {1}{2}}(-\sigma _{xz}+\pi \sigma _{yz})&-{\cfrac {2}{3}}\sigma _{yz}-{\cfrac {\pi }{2}}\sigma _{xz}&-{\cfrac {7}{6}}\sigma _{zz}\end{bmatrix}}}$

## Problem 2: Hyperelastic Pinched Cylinder Problem

Read the following paper on shells:
Buchter, N., Ramm, E., and Roehl, D., 1994, "Three-dimensional extension of non-linear shell formulation based on the enhanced assumed strain concept," Int. J. Numer. Meth. Engng., 37, pp. 2551-2568.
Answer the following questions:

### Solution
#### Part 1

What do the authors mean by "enhanced assumed strain"?
See Wikipedia article on [w:Enhanced assumed strain|enhanced assumed strain]] .

#### Part 2

Example 8.2 (and Figures 3 and 4 and Table III) in the paper discusses the simulation of a hyperelastic cylinder. Perform the same simulation using ANSYS for a shell thickness of 0.2 cm. Use shell elements and the Neo-Hookean hyperelastic material model that ANSYS provides.
The following material properties are used:
$E=16800$ kN/cm$^{2}$, $\nu =0.4$, $\mu =6000$ kN/cm$^{2}$, and $d=2/K=7.14\times 10^{-5}$ cm$^{2}$/kN. Symmetry is used and only half of the model is meshed. At the support, the model is constraint in all directions. The load of 36 kN is applied on the top of the cylinder (Fig~2. ANSYS input listing is shown Fig~6 of Appendix.
* Figure 2. Meshed model

#### Part 3

Compare the total load needed to achieve an edge displacement of 16 cm with the results given in Table III. Comment on your results.
The plot of force vs. edge displacement (vertical) and the deformed model are shown in Fig 3. From the plot, one sees that a load of 35.1 kN is required to deform the edge by 16 cm. This is less than 1% difference compared to the result given in the paper using 7-parameter shell theory.
* Figure 3. Left: Force vs. edge displacement. Right: Deformed model.

## Problem 3: Elastic-Plastic Punch Indentation

Read the following paper on elastic-viscoplastic FEA:
Rouainia, M. and Peric, D., 1998, "A computational model for elasto-viscoplastic solids at finite strain with reference to thin shell applications," Int. J. Numer. Meth. Engng., 42, pp. 289-311.
Answer the following questions:

### Solution
#### Part 1

Example 5.4 of the paper shows a simulation of the deformation of a thin sheet by a square punch. Perform the same simulation for 6061-T6 aluminum. Assume linear isotropic hardening and no rate dependence.
The following data are used for the thin plate:$E=70$ GPa, $\nu =0.3$, $\sigma _{Y}=240$ Mpa, $\sigma _{t}=7800$ Mpa, see Fig 4 for the stress-strain data. Symmetry is used and only half of the model is meshed (Fig 4). At the support, the model is constrained in all directions. A plastic finite strained shell (SHELL43) is chosen for this simulation.
The following material properties are used for the punch and die:$E=100\times 10^{6}$ GPa, $\nu =0.3$. The value of Young's modulus is arbitrary chosen so long as it is high enough to remain rigid during the simulation.
The meshed model is illustrated in Fig 4. A load of 10 kN is applied on the top of the punch. Load steps are split into two steps. First load step the punch is moved close to the plate to activate the contact elements (CONTAC49). The second load step, 30 kN is applied on top of the punch. ANSYS input listing is shown Fig 7 of the Appendix.
* Figure 4. Left: Bi-linear stress-strain data. Right: Meshed model.

#### Part 2

Draw a plot of the punch force vs. punch travel and compare your result with the results shown in Figure 13 of the paper (qualitative comparison only).
The punch force vs. punch travel and the plot of the deformation at the final load step is shown in Fig 5. The curve displays similar pattern as those shown in the paper.
* Figure 5. Left: Punch force vs. punch travel. Right: Sketch of deformation at final load step.

### Appendix

* Figure 6. ANSYS input listing for Problem 3.
* Figure 7. ANSYS input listing for Problem 3.