Conservation of Energy First Principle of Thermodynamics

30 The first principle of thermodynamics relates the work done on a (closed) system and the heat transfer into the system to the change in energy of the system. We shall assume that the only energy transfers to the system are by mechanical work done on the system by surface traction and body forces, by heat transfer through the boundary.

6.4.1 Spatial Gradient of the Velocity

31 We define L as the spatial gradient of the velocity and in turn this gradient can be decomposed into a symmetric rate of deformation tensor D (or stretching tensor) and a skew-symmeteric tensor W called the spin tensor or vorticity tensor1.

this term will be used in the derivation of the first principle.

6.4.2 First Principle

32 If mechanical quantities only are considered, the principle of conservation of energy for the continuum may be derived directly from the equation of motion given by Eq. 6.24. This is accomplished by taking the integral over the volume V of the scalar product between Eq. 6.24 and the velocity v^.

1 Note similarity with Eq. 4.111-b.

If we consider the right hand side

v pv didV = dtj v 2 ^dV = dtj v 2 ^ dV = ~KK (6-40)

which represents the time rate of change of the kinetic energy K in the continuum.

33 Also we have ViTji,j = (ViTji)jj — Vi,jTji and from Eq. 6.37 we have Vi,j = Lij + Wij. It can be shown that since Wij is skew-symmetric, and T is symmetric, that TijWij = 0, and thus TijLj = TjDij. TD is called the stress power.

34 If we consider thermal processes, the rate of increase of total heat into the continuum is given by

Q has the dimension of power, that is ML2T-3, and the SI unit is the Watt (W). q is the heat flux per unit area by conduction, its dimension is MT-3 and the corresponding SI unit is Wm-2. Finally, r is the radiant heat constant per unit mass, its dimension is MT-3L-4 and the corresponding SI unit is Wm-6.

35 We thus have dK

dK + J Dij Tij dV = J V Tji),j dV + j pvibidV + Q (6.42)

36 We next convert the first integral on the right hand side to a surface integral by the divergence theorem (fv V-vdV = fs v.ndS) and since ti = Tjnj we obtain

dK

f

r

f pVibidV + Q (6.43)

dt

/ Dij Tij dV =

/ VitidS +

v

J S J

v

dK dU

dW

(6.44)

dt dt

~dT + Q

this equation relates the time rate of change of total mechanical energy of the continuum on the left side to the rate of work done by the surface and body forces on the right hand side.

37 If both mechanical and non mechanical energies are to be considered, the first principle states that the time rate of change of the kinetic plus the internal energy is equal to the sum of the rate of work plus all other energies supplied to, or removed from the continuum per unit time (heat, chemical, electromagnetic, etc.).

38 For a thermomechanical continuum, it is customary to express the time rate of change of internal energy by the integral expression dU d ¡' .

where u is the internal energy per unit mass or specific internal energy. We note that U appears only as a differential in the first principle, hence if we really need to evaluate this quantity, we need to have a reference value for which U will be null. The dimension of U is one of energy dim U = ML2T-2, and the SI unit is the Joule, similarly dim u = L2T-2 with the SI unit of Joule/Kg.

39 In terms of energy integrals, the first principle can be rewritten as

Rate of increae Exchange Source Source Exchange dtt / 2PViVidV + dt i pudV = f tiVidS + f pVibidV + f prdV — ( qinidS (6.46)

dK dU dW Q

dt dt dt we apply Gauss theorem to convert the surface integral, collect terms and use the fact that dV is arbitrary to obtain

du

(6.47)

= T:D + pr -

or

du

= Tij Dij + pr

_ dj dxj

(6.48)

40 This equation expresses the rate of change of internal energy as the sum of the stress power plus the heat added to the continuum.

41 In ideal elasticity, heat transfer is considered insignificant, and all of the input work is assumed converted into internal energy in the form of recoverable stored elastic strain energy, which can be recovered as work when the body is unloaded.

42 In general, however, the major part of the input work into a deforming material is not recoverably stored, but dissipated by the deformation process causing an increase in the body's temperature and eventually being conducted away as heat.

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