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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