Lab 2, Section 2: Solutions

Steps 1-8: Uniaxial extension of a unit cube

\(\theta=0\) degrees (Steps 1-2):

\[ \begin{align}\begin{aligned}\begin{split}\boldsymbol{E_{ref}} = \boldsymbol{E_{fib}} = \begin{bmatrix} 0.6250 & 0 & 0 \\ 0 & -0.1667 & 0 \\ 0 & 0 & -0.1667 \end{bmatrix}\end{split}\\\begin{split}\boldsymbol{T_{ref}} = \boldsymbol{T_{fib}} = \begin{bmatrix} 440.5 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}\end{split}\end{aligned}\end{align} \]

\(\theta=30\) degrees (Steps 3-7):

\[ \begin{align}\begin{aligned}\begin{split}\boldsymbol{E_{ref}} = \begin{bmatrix} 0.6250 & 0 & 0 \\ 0 & -0.1667 & 0 \\ 0 & 0 & -0.1667 \end{bmatrix}\end{split}\\\begin{split}\boldsymbol{E_{fib}} = \begin{bmatrix} 0.4271 & -0.3428 & 0 \\ -0.3428 & 0.0313 & 0 \\ 0 & 0 & -0.1667 \end{bmatrix}\end{split}\\\begin{split}\boldsymbol{T_{ref}} = \begin{bmatrix} 440.5 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}\end{split}\\\begin{split}\boldsymbol{T_{fib}} = \begin{bmatrix} 330.345 & -190.725 & 0 \\ -190.725 & 110.115 & 0 \\ 0 & 0 & 0 \end{bmatrix}\end{split}\end{aligned}\end{align} \]

\(\theta=45\) degrees (Step 8):

\[ \begin{align}\begin{aligned}\begin{split}\boldsymbol{E_{ref}} = \begin{bmatrix} 0.6250 & 0 & 0 \\ 0 & -0.1667 & 0 \\ 0 & 0 & -0.1667 \end{bmatrix}\end{split}\\\begin{split}\boldsymbol{E_{fib}} = \begin{bmatrix} 0.2292 & -0.3958 & 0 \\ -0.3958 & 0.2292 & 0 \\ 0 & 0 & -0.1667 \end{bmatrix}\end{split}\\\begin{split}\boldsymbol{T_{ref}} = \begin{bmatrix} 440.5 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}\end{split}\\\begin{split}\boldsymbol{T_{fib}} = \begin{bmatrix} 220.2 & -220.2 & 0 \\ -220.2 & 220.2 & 0 \\ 0 & 0 & 0 \end{bmatrix}\end{split}\end{aligned}\end{align} \]

\(\theta=90\) degrees (Step 8):

\[ \begin{align}\begin{aligned}\begin{split}\boldsymbol{E_{ref}} = \begin{bmatrix} 0.6250 & 0 & 0 \\ 0 & -0.1667 & 0 \\ 0 & 0 & -0.1667 \end{bmatrix}\end{split}\\\begin{split}\boldsymbol{E_{fib}} = \begin{bmatrix} -0.1667 & 0 & 0 \\ 0 & 0.6250 & 0 \\ 0 & 0 & -0.1667 \end{bmatrix}\end{split}\\\begin{split}\boldsymbol{T_{ref}} = \begin{bmatrix} 440.5 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}\end{split}\\\begin{split}\boldsymbol{T_{fib}} = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 440.5 & 0 \\ 0 & 0 & 0 \end{bmatrix}\end{split}\end{aligned}\end{align} \]

Step 21: Equi-biaxial extension of a unit cube

\(\theta=45\) degrees:

\[ \begin{align}\begin{aligned}\begin{split}\boldsymbol{E_{ref}} = \boldsymbol{E_{fib}} = \begin{bmatrix} 0.2812 & 0 & 0 \\ 0 & 0.2812 & 0 \\ 0 & 0 & -0.2952 \end{bmatrix}\end{split}\\\begin{split}\boldsymbol{T_{ref}} = \boldsymbol{T_{fib}} = \begin{bmatrix} 10.95 & 0 & 0 \\ 0 & 10.95 & 0 \\ 0 & 0 & 0 \end{bmatrix}\end{split}\end{aligned}\end{align} \]

\(\theta=90\) degrees:

\[ \begin{align}\begin{aligned}\begin{split}\boldsymbol{E_{ref}} = \boldsymbol{E_{fib}} = \begin{bmatrix} 0.2812 & 0 & 0 \\ 0 & 0.2812 & 0 \\ 0 & 0 & -0.2952 \end{bmatrix}\end{split}\\\begin{split}\boldsymbol{T_{ref}} = \boldsymbol{T_{fib}} = \begin{bmatrix} 10.95 & 0 & 0 \\ 0 & 10.95 & 0 \\ 0 & 0 & 0 \end{bmatrix}\end{split}\end{aligned}\end{align} \]