ABSTRACT
Topics addressed in this chapter include linear conductive heat transfer, linear thermoelasticity, incompressible elastic materials, elastic torsion, and buckling.
The governing equation for conductive heat transfer without heat sources in an isotropic medium is
kr2T ¼ rce _T (11:1)
in which T is the (absolute) temperature, k is the thermal conductivity, and ce is the coefficient of specific heat at constant strain. We invoke the interpolation model T(t)T0¼wTT (x)FTu(t) in which u(t) is the vector of nodal temperatures (minus T0), while wTT (x) and FT are the thermal counterparts of w
T(x) and F in mechanical fields. Also application of the gradient leads to a relation of the formrT¼bTT FT u(t), and the finite element equation assumes the form
KTuþMT _u ¼ q(T) (11:2)
KT ¼ ð FTTbTkb
T TFT dV , MT ¼
ð FTTwTrcew
This equation is parabolic (first order in the time rates), and implies that the temperature changes occur immediately at all points in the domain, but at smaller initial rates away from where the heat is added. This contrasts with the hyperbolic (second order in time rates) solid mechanics equations, in which information propagates into the unperturbed medium as finite velocity waves, and in which oscillatory response occurs in response to a perturbation.
