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

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
    $x({\boldsymbol {\xi }},t)=\sum _{i=1}^{3}x_{i}(t)~N_{i}({\boldsymbol {\xi }})~;\qquad y({\boldsymbol {\xi }},t)=\sum _{i=1}^{3}y_{i}(t)~N_{i}({\boldsymbol {\xi }})$
* Write down expressions for $N_{1}$, $N_{2}$, and $N_{3}$ in terms of the initial configuration ($X,Y$) ?
* Derive expressions for the deformation gradient and the Jacobian determinant for the element as functions of time.
* What are the values of $a$ and $b$ for which the motion is isochoric?
* For which values of $a$ and $b$ do we get invalid motions?
* Derive the expression for the Green (Lagrangian) strain tensor for the element as a function of time.
* Derive an expression for the velocity gradient as a function of time.
* Compute the rate of deformation tensor and the spin tensor.
* Assume that $a=1$ and $b=2$. Sketch the undeformed configuration and the deformed configuration at $t=1$ and $t=2$. Draw both the deformed and undeformed configurations on the same plot and label.
* Compute the polar decomposition of the deformation gradient with the above values of $a$ and $b$.
* Assume an isotropic, hypoelastic constitutive equation for the material of the element. Compute the material time derivative of the Cauchy stress using (a) the Jaumann rate and (b) the Truesdell rate.

## 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:
* What do the authors mean by "enhanced assumed strain" ?
* 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.
* Compare the total load needed to achieve an edge displacement of 16 cm with the results given in Table III. Comment on your results.

## 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:
* 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.
* 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).