Balance of Equations and Unknowns

58 In the preceding sections several equations and unknowns were introduced. Let us count them. for both the coupled and uncoupled cases.

Coupled

Uncoupled

dP + p ävi — o dt^ P dxi U

Continuity Equation

1

1

dj + Pbi — p ddt

Equation of motion

3

3

P du — Tn Dn + Pr - Hj

Energy equation

1

Total number of equations

5

4

59 Assuming that the body forces bi and distributed heat sources r are prescribed, then we have the following 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 js > ^ — - div q which governs entropy production must hold. P

60 We thus need an additional 16 — 5 = 11 additional equations to make the system determinate. These will be later on supplied by:

6

constitutive equations

3

temperature heat conduction

2

thermodynamic equations of state

11

Total number of additional equations

61 The next chapter will thus discuss constitutive relations, and a subsequent one will separately discuss thermodynamic equations of state.

62 We note that for the uncoupled case

1. The energy equation is essentially the integral of the equation of motion.

2. The 6 missing equations will be entirely supplied by the constitutive equations.

3. The temperature field is regarded as known, or at most, the heat-conduction problem must be solved separately and independently from the mechanical problem.

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