## Updated Balance of Equations and Unknowns

76 In light of the new equations introduced in this chapter, it would be appropriate to revisit our balance of equations and unknowns.

Coupled |
Uncoupled | ||

dp + p % = 0 |
Continuity Equation |
1 |
1 |

dj + pbi = p it |
Equation of motion |
3 |
3 |

P dU = Tj Du + Pr - |
Energy equation |
1 | |

T = XIE + 2/xE |
Hooke's Law |
6 |
6 |

q = -DV^ |
Heat Equation (Fourrier) |
3 | |

0 = 0(s, V); Tj = Tj (s, V) |
Equations of state |
2 | |

Total number of equations |
16 |
10 |

and we repeat our list of unknowns

Coupled |
Uncoupled | ||

Density |
P |
1 |
1 |

Velocity (or displacement) |
Vi (Ui) |
3 |
3 |

Stress components |
T- ■ |
6 |
6 |

Heat flux components |
qi |
3 |
- |

Specific internal energy |
u |
1 |
- |

Entropy density |
s |
1 |
- |

Absolute temperature |
0 |
1 |
- |

Total number of unknowns |
16 |
10 |

and in addition the Clausius-Duhem inequality dt > — 1 div |L which governs entropy production must hold.

77 Hence we now have as many equations as unknowns and are (almost) ready to pose and solve problems in continuum mechanics.

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