7.2 The Governing Equations

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In the present derivation we will allow for plastic processes, but we assume small strains,

that is, we start from a free energy potential of the form in Equation (5.11):

_ = _(_ − _p, α, θ,∇θ,X). (7.1)

Furthermore, rectangular coordinates are assumed throughout. Of course, the conservation

laws still apply. In particular, the Clausius–Duhem inequality leads to (cf Chapter 5)

σ = ρ

ρ0

∂_

∂_e (7.2)

η = − 1

ρ0

∂_

∂θ

(7.3)

The Finite Element Method for Three-dimensional Thermomechanical Applications Guido Dhondt

2004 John Wiley & Sons, Ltd ISBN: 0-470-85752-8

306 HEAT TRANSFER

qi = −∂_

∂α

(7.4)

∂_

∂∇θ

= 0 (7.5)

where qi stands for the internal dynamic variables. The index ‘i’ was introduced to avoid

confusion with the heat flux q. Introducing a reference temperature θref, we define the

relative temperature

T := θ − θref. (7.6)

T is assumed to be small compared to θref. We now expand _ as a function of T as follows

(_ does not depend on ∇θ because of Equation (7.5)):

_(_e, α, θ,X) = ρ0(X)ψ0(_e, α,X) − ρ0(X)η0(_e,X)T − _ρ0(X)c(_e, θ,X)

2θref

_

T 2.

(7.7)

This is an equality, not an approximation: notice that c is a function of the temperature θ. It

is assumed that the dependence on α does not depend on the temperature (only ψ0 contains

α). Applying Equations (7.2) to (7.4) and keeping the linear terms only (T is assumed to

be small) leads to

σ = ρ

∂ψ0

∂_e

− ρ

∂η0

∂_e T + O(T 2) (7.8)

η = η0 + cT

θref

(7.9)

qi = −ρ0

∂ψ0

∂α

(7.10)

where ρ0(X), ψ0(_e, α,X), η0(_e, X) and c(_e, θ, X), as in Equation (7.7). Equation (7.8)

splits the stresses into a mechanical part and a thermal part.

The internal energy satisfies (cf Equation (1.387):

ε = _

ρ0

+ θη. (7.11)

Substitution of Equations (7.7) and (7.9) into Equation (7.11) yields

ε = ψ0 − η0T + θη0 + cT

θref

θ + O(T 2) (7.12)

which can be further simplified to

ε = ψ0(_e, α,X) + η0(_e,X)θref + c(_e, θ,X)T + O(T 2). (7.13)

The conservation of energy requires (Equation (1.355), spatial form)

ρ˙ε = ˙_ : σ − qk

,k

+ ρh (7.14)

HEAT TRANSFER 307

which yields after the use of Equation (7.13)

ρ

∂ψ0

∂_e : ˙_e + ρ

∂ψ0

∂α

: ˙α + ρ

∂η0

∂_e : ˙_eθref + ρ

∂c

∂_e : ˙_ eT

+ ρ

∂c

∂T

˙ T T + ρc ˙ T − (˙_ e + ˙_p) : σ + qk

,k

− ρh = 0. (7.15)

The first term in Equation (7.15) is a linear approximation to σ : _e (cf Equation (7.8)),

the second term corresponds to −qi : ˙α (cf Equation (7.10)) and the fourth and fifth terms

are quadratic (T and ˙_e are both small). Accordingly, Equation (7.15) reduces to

ρc ˙ T = −qk

,k

+ ρh + σ : ˙_ p + qi : ˙α − β : ˙_eθref (7.16)

where

β := ρ

∂η0

∂_e (7.17)

is the stress reduction per temperature increase (cf Equations (7.8) and (1.413)). Equation

(7.16) expresses that a temperature increase can result from heat flux, heat sources,

plastic dissipation, internal-variable dissipation and the work rate of the thermal stresses or

any combination. The last three terms depend on the deformation and embody the influence

of the deformation (mechanical action) on the temperature. The conservation of energy in

the form of Equation (7.16) is the governing equation in heat-transfer calculations.

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