Homework #12
-
Start with the following equation from the
fluid mechanics page
\[
\rho \, \left( {\partial {\bf v} \over \partial t} +
{\bf v} \cdot \nabla {\bf v} \right) =
-\nabla P + 2 \mu \nabla \cdot {\bf D}' + \rho \, {\bf f}
\]
and use tensor notation to show how to get to this equation,
which occurs farther down the page.
\[
\rho \, \left( {\partial {\bf v} \over \partial t} +
{\bf v} \cdot \nabla {\bf v} \right) =
-\nabla P + \mu \nabla^2 {\bf v}
+ \rho \, {\bf f}
\]
-
The (made up) test data below is for tension tests of a
rubber sample at two temperatures. Propose a Helmholtz
function in terms of strain and temperature and show that
it reproduces the measured data. (Note - The Helmholtz
function must be a function of temperature in Kelvin,
not Celsius.)
| 27°C | 77°C |
Strain | Stress (MPa) | Stress (MPa) |
0 | 0 | 0 |
0.05 | 0.3 | 0.2 |
0.10 | 0.7 | 0.4 |
0.15 | 1.0 | 0.7 |
0.20 | 1.5 | 0.8 |
0.25 | 1.8 | 1.1 |
0.30 | 2.5 | 1.6 |
0.35 | 3.0 | 1.9 |
0.40 | 3.5 | 2.4 |
0.45 | 4.2 | 2.9 |
0.50 | 4.9 | 3.3 |
0.55 | 5.7 | 3.8 |
0.60 | 6.7 | 4.3 |
0.65 | 7.5 | 4.9 |
0.70 | 8.5 | 5.5 |
0.75 | 9.4 | 6.2 |
0.80 | 10.4 | 6.9 |
0.85 | 11.5 | 7.6 |
0.90 | 12.7 | 8.4 |