Homework #12



  1. 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.

    \[ \rho \, \left( {\partial {\bf v} \over \partial t} + {\bf v} \cdot \nabla {\bf v} \right) = -\nabla P + \mu \nabla^2 {\bf v} + \rho \, {\bf f} \]


  2. 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°C77°C
    StrainStress (MPa)Stress (MPa)
    000
    0.050.30.2
    0.100.70.4
    0.151.00.7
    0.201.50.8
    0.251.81.1
    0.302.51.6
    0.353.01.9
    0.403.52.4
    0.454.22.9
    0.504.93.3
    0.555.73.8
    0.606.74.3
    0.657.54.9
    0.708.55.5
    0.759.46.2
    0.8010.46.9
    0.8511.57.6
    0.9012.78.4