### 5.2 Transient thermal distortions

In the quasistatic regime, the time scale for temperature evolution is clearly too long to generate inertial
effects in the material. Thus, the elastodynamic equations reduce to elastic. The equilibrium equations are
therefore unchanged with respect to Equation (3.91). The displacement vector is unchanged in form with
respect to the static case, except that the time enters as an evolution parameter through temperature. The
temperature field being, as usual,
The generic thermoelastic longitudinal displacement is of the form

We are interested in the displacement of the reflecting surface, and the general solution gives
where the function is a special solution of
and where the notation of Equation (3.148) has been employed.

#### 5.2.1 Case of coating absorption

In the case of a heat source on the coating, we have found the temperature field from Equation (5.33), so
that we have

and consequently
with
so that finally
The contribution to curvature of the Saint-Venant correction reduces to
with the time dependent curvature
See in Figures 44, 45, and 46, the time evolution of the surface deformation under a constant power flux.
The dashed curves corresponding to the steady state are computed with Equations (3.120) and (3.122).
Using our averaging technique, we can compute the transient curvature radius for the three considered
examples (Figures 47, 48, and 49).
#### 5.2.2 Case of bulk absorption

The temperature field defined by Equation (5.37) gives

so that
with
yielding, finally,
Due to the symmetry of the temperature field in and to the resulting symmetry of the stress field
, the contribution to curvature of the Saint-Venant correction (mean torque) is zero. See in
Figures 50, 51, and 52 the time evolution of the distorted surface for our three examples. The curvature
evolution is plotted in Figures 53,54, and 55.